Voting forms the most important tool for arriving at a decision in any institution. The changing needs of the civilization currently demands a practical yet secure electronic voting system, but any flaw related to the applied voting technology can lead to tampering of the results with the malicious outcomes. Currently, blockchain technology due to its transparent structure forms an emerging area of investigation for the development of voting systems with a far greater security. However, various apprehensions are yet to be conclusively resolved before using blockchain in high stakes elections. Other than this, the blockchain based voting systems are vulnerable to possible attacks by upcoming noisy intermediate scale quantum (NISQ) computer. To circumvent, most of these limitations, in this work, we propose an anonymous voting scheme based on quantum assisted blockchain by enhancing the advantages offered by blockchain with the quantum resources such as quantum random number generators and quantum key distribution. The purposed scheme is shown to satisfy the requirements of a good voting scheme. Further, the voting scheme is auditable and can be implemented using the currently available technology.
Smart contracts are cryptographic protocols that are enforced without a judiciary. Smart contracts are used occasionally in Bitcoin and are prevalent in Ethereum. Public quantum money improves upon cash we use today, yet the current constructions do not enable smart contracts. In this work, we define and introduce quantum payment schemes, and show how to implement prudent contracts -- a non-trivial subset of the functionality that a network such as Ethereum provides. Examples discussed include: multi-signature wallets in which funds can be spent by any 2-out-of-3 owners; restricted accounts that can send funds only to designated destinations; and "colored coins" that can represent stocks that can be freely traded, and their owner would receive dividends. Our approach is not as universal as the one used in Ethereum since we do not reach a consensus regarding the state of a ledger. We call our proposal prudent contracts to reflect this. The main building block is either quantum tokens for digital signatures (Ben-David and Sattath QCrypt'17, Coladangelo et al. Crypto'21), semi-quantum tokens for digital signatures (Shmueli'22) or one-shot signatures (Amos et al. STOC'20). The solution has all the benefits of public quantum money: no mining is necessary, and the security model is standard (e.g., it is not susceptible to 51\% attacks, as in Bitcoin). Our one-shot signature construction can be used to upgrade the Bitcoin network to a quantum payment scheme. Notable advantages of this approach are: transactions are locally verifiable and without latency, the throughput is unbounded, and most importantly, it would remove the need for Bitcoin mining. Our approach requires a universal large-scale quantum computer and long-term quantum memory; hence we do not expect it to be implementable in the next few years.
Despite all the progress in quantum technologies over the last decade, there is still a dearth of practical applications for quantum computers with a small number of noisy qubits. The effort to show quantum supremacy has been largely focused on demonstrating computations that cannot be accomplished on a classical computer at all, a difficult and controversial target. Quantum advantage (a speedup over classical computers) is a more practical milestone for today's modest quantum processors. In this work, we proposed a scheme for quantum-computer compatible proof of work (cryptographic mechanism used in Bitcoin mining) and verified it on a 4-qubit superconducting quantum node.
Marco Pistoia, Omar Amer, Monik R. Behera, Joseph A. Dolphin · 19 authors
This article describes experimental research studies conducted towards understanding the implementation aspects of high-capacity quantum-secured optical channels in mission-critical metro-scale operational environments using Quantum Key Distribution (QKD) technology. To the best of our knowledge, this is the first time that an 800 Gbps quantum-secured optical channel -- along with several other Dense Wavelength Division Multiplexed (DWDM) channels on the C-band and multiplexed with the QKD channel on the O-band -- was established at distances up to 100 km, with secret key-rates relevant for practical industry use cases. In addition, during the course of these trials, transporting a blockchain application over this established channel was utilized as a demonstration of securing a financial transaction in transit over a quantum-secured optical channel. The findings of this research pave the way towards the deployment of QKD-secured optical channels in high-capacity, metro-scale, mission-critical operational environments, such as Inter-Data Center Interconnects.
Quantum entangled states have shown distinguished features beyond any classical state. Many methods like quantum state tomography have been presented to verify entanglement. In this work, we aim to identify unknown entanglements with partial information of the state space by developing a nonlinear entanglement witness. The witness consists of a generalized Greenberger-Horne-Zeilinger-like paradox expressed by Pauli observables, and a nonlinear inequality expressed by density matrix elements. First, we verify unknown bipartite entanglements and study the robustness of entanglement witnesses against the white noise. Second, we generalize such a verification to unknown multipartite entangled states, including the Greenberger-Horne-Zeilinger-type states and the cluster states under local channel operations. Third, we give a quantum-information application related to the quantum zero-knowledge proof. Our results provide a useful method in verifying universal quantum computation resources with robustness against white noises. Our work is applicable to detect unknown entanglement without the state tomography.
Zero-knowledge proof (ZKP) is a fundamental cryptographic primitive that allows a prover to convince a verifier of the validity of a statement without leaking any further information. As an efficient variant of ZKP, noninteractive zero-knowledge proof (NIZKP) adopting the Fiat-Shamir heuristic is essential to a wide spectrum of applications, such as federated learning, blockchain, and social networks. However, the heuristic is typically built upon the random oracle model that makes ideal assumptions about hash functions, which does not hold in reality and thus undermines the security of the protocol. Here, we present a quantum solution to the problem. Instead of resorting to a random oracle model, we implement a quantum randomness service. This service generates random numbers certified by the loophole-free Bell test and delivers them with postquantum cryptography (PQC) authentication. By employing this service, we conceive and implement NIZKP of the three-coloring problem. By bridging together three prominent research themes, quantum nonlocality, PQC, and ZKP, we anticipate this work to inspire more innovative applications that combine quantum information science and the cryptography field.
The advancement of quantum computers undermines the security of classical blockchain, necessitating either a post-quantum upgrade of the existing architecture or creation of an inherently quantum blockchain. Here we propose a practically realizable model of a fully quantum blockchain based on a generalized Gram-Schmidt procedure utilizing dimensional lifting. In this model, information of transactions stored in a multi-qubit state are subsequently encoded using the generalized Gram-Schmidt process. The chain is generated as a result of the reliance of orthogonalized state on the sequence of states preceding it. Various forking scenarios and their countermeasures are considered for the proposed model. It is shown to be secure even against quantum computing attacks using the no-cloning theorem and non-democratic nature of Generalized Gram-Schmidt orthogonalization. Finally, we outline a framework for a quantum token built on the same architecture as our blockchain.
Our aim is to determine conditions for quantum computing technology to give rise to security risks associated with quantum Bitcoin mining. Specifically, we determine the speed and energy efficiency a quantum computer needs to offer an advantage over classical mining. We analyze the setting in which the Bitcoin network is entirely classical except for a single quantum miner who has small hash rate compared to that of the network. We develop a closed-form approximation for the probability that the quantum miner successfully mines a block, with this probability dependent on the number of Grover iterations the quantum miner applies before making a measurement. Next, we show that, for a quantum miner that is "peaceful", this success probability is maximized if the quantum miner applies Grover iterations for 16 minutes before measuring, which is surprising as the network mines blocks every 10 minutes on average. Using this optimal mining procedure, we show that the quantum miner outperforms a classical computer in efficiency (cost per block) if the condition $Q < Crb$ is satisfied, where $Q$ is the cost of a Grover iteration, $C$ is the cost of a classical hash, $r$ is the quantum miner's speed in Grover iterations per second, and $b$ is a factor that attains its maximum if the quantum miner uses our optimal mining procedure. This condition lays the foundation for determining when quantum mining, and the known security risks associated with it, will arise.
In known constructions of classical zero-knowledge protocols for NP, either of zero-knowledge or soundness holds only against computationally bounded adversaries. Indeed, achieving both statistical zero-knowledge and statistical soundness at the same time with classical verifier is impossible for NP unless the polynomial-time hierarchy collapses, and it is also believed to be impossible even with a quantum verifier. In this work, we introduce a novel compromise, which we call the certified everlasting zero-knowledge proof for QMA. It is a computational zero-knowledge proof for QMA, but the verifier issues a classical certificate that shows that the verifier has deleted its quantum information. If the certificate is valid, even unbounded malicious verifier can no longer learn anything beyond the validity of the statement. We construct a certified everlasting zero-knowledge proof for QMA. For the construction, we introduce a new quantum cryptographic primitive, which we call commitment with statistical binding and certified everlasting hiding, where the hiding property becomes statistical once the receiver has issued a valid certificate that shows that the receiver has deleted the committed information. We construct commitment with statistical binding and certified everlasting hiding from quantum encryption with certified deletion by Broadbent and Islam [TCC 2020] (in a black box way), and then combine it with the quantum sigma-protocol for QMA by Broadbent and Grilo [FOCS 2020] to construct the certified everlasting zero-knowledge proof for QMA. Our constructions are secure in the quantum random oracle model. Commitment with statistical binding and certified everlasting hiding itself is of independent interest, and there will be many other useful applications beyond zero-knowledge.
Mark Webber, Vincent Elfving, Sebastian Weidt, Winfried K. Hensinger
We investigate how hardware specifications can impact the final run time and the required number of physical qubits to achieve a quantum advantage in the fault tolerant regime. Within a particular time frame, both the code cycle time and the number of achievable physical qubits may vary by orders of magnitude between different quantum hardware designs. We start with logical resource requirements corresponding to a quantum advantage for a particular chemistry application, simulating the FeMoco molecule, and explore to what extent slower code cycle times can be mitigated by using additional qubits. We show that in certain situations architectures with considerably slower code cycle times will still be able to reach desirable run times, provided enough physical qubits are available. We utilize various space and time optimization strategies that have been previously considered within the field of error-correcting surface codes. In particular, we compare two distinct methods of parallelization, Game of Surface Code's Units, and AutoCCZ factories, both of which enable one to incrementally speed up the computation until the reaction limited rate is reached. Finally we calculate the number of physical qubits which would be required to break the 256 bit elliptic curve encryption of keys in the Bitcoin network, within the small available time frame in which it would actually pose a threat to do so. It would require approximately 317 million physical qubits to break the encryption within one hour using the surface code, a code cycle time of 1 $ μs$, a reaction time of 10 $ μs$, and physical gate error of $10^{-3}$. To break the encryption instead within one day it would require 13 million physical qubits.
The advent of quantum computing threatens blockchain protocols and networks because they utilize non-quantum resistant cryptographic algorithms. When quantum computers become robust enough to run Shor's algorithm on a large scale, the most used asymmetric algorithms, utilized for digital signatures and message encryption, such as RSA, (EC)DSA, and (EC)DH, will be no longer secure. Quantum computers will be able to break them within a short period of time. Similarly, Grover's algorithm concedes a quadratic advantage for mining blocks in certain consensus protocols such as proof of work. Today, there are hundreds of billions of dollars denominated in cryptocurrencies and other digital assets that rely on blockchain ledgers as well as thousands of blockchain-based applications storing value in blockchain networks. Cryptocurrencies and blockchain-based applications require solutions that guarantee quantum resistance in order to preserve the integrity of data and assets in these public and immutable ledgers. The quantum threat and some potential solutions are well understood and presented in the literature. However, most proposals are theoretical, require large QKD networks, or propose new quantum-resistant blockchain networks to be built from scratch. Our work, which is presented in this paper, is pioneer in proposing an end-to-end framework for post-quantum blockchain networks that can be applied to existing blockchain to achieve quantum-resistance. We have developed an open-source implementation in an Ethereum-based (i.e., EVM compatible) network that can be extended to other existing blockchains. For the implementation we have (i) used quantum entropy to generate post-quantum key pairs, (ii) established post-quantum TLS connections and X.509 certificates to secure the exchange of information between blockchain nodes over the internet without needing a large QKD network, (iii) introduced a post-quantum second signature in transactions using Falcon-512 post-quantum keys, and (iv) developed the first on-chain verification of post-quantum signatures using three different mechanisms that are compared and analyzed: Solidity smart-contracts run by the validators for each transaction, modified EVM Opcode, and precompiled smart contracts.
Dan A. Bard, Joseph J. Kearney, Carlos A. Pérez-Delgado
Proof-of-Work (PoW) is a fundamental underlying technology behind most major blockchain cryptocurrencies. It has been previously pointed out that quantum devices provide a computational advantage in performing PoW in the context of Bitcoin. Here we make the case that this quantum advantage extends not only to all existing PoW mechanisms, but to any possible PoW as well. This has strong consequences regarding both quantum-based attacks on the integrity of the entirety of the blockchain, as well as more legitimate uses of quantum computation for the purpose of mining Bitcoin and other cryptocurrencies. For the first case, we estimate when these quantum attacks will become feasible, for various cryptocurrencies, and discuss the impact of such attacks. For the latter, we derive a precise formula to calculate the economic incentive for switching to quantum-based cryptocurrency miners. Using this formula, we analyze several test scenarios, and conclude that investing in quantum hardware for cryptocurrency mining has the potential to pay off immensely.
Quantum computation represents a threat to many cryptographic protocols in operation today. It has been estimated that by 2035, there will exist a quantum computer capable of breaking the vital cryptographic scheme RSA2048. Blockchain technologies rely on cryptographic protocols for many of their essential sub-routines. Some of these protocols, but not all, are open to quantum attacks. Here we analyze the major blockchain-based cryptocurrencies deployed today -- including Bitcoin, Ethereum, Litecoin and ZCash, and determine their risk exposure to quantum attacks. We finish with a comparative analysis of the studied cryptocurrencies and their underlying blockchain technologies and their relative levels of vulnerability to quantum attacks.
We propose the first generalization of the famous Non-Interactive Zero-Knowledge (NIZK) proofs to quantum languages (NIZKoQS) and we provide a protocol to prove advanced properties on a received quantum state non-destructively and non-interactively (a single message being sent from the prover to the verifier). In our second orthogonal contribution, we improve the costly Remote State Preparation protocols [Cojocaru et al. 2019 ; Gheorghiu and Vidick 2019 ] that can classically fake a quantum channel (this is at the heart of our NIZKoQS protocol) by showing how to create a multi-qubit state from a single superposition. Finally, we generalize these results to a multi-party setting and prove that multiple parties can anonymously distribute a GHZ state in such a way that only participants knowing a secret credential can share this state, which could have applications to quantum anonymous transmission, quantum secret sharing, quantum onion routing and more.
We propose three constructions of classically verifiable non-interactive zero-knowledge proofs and arguments (CV-NIZK) for QMA in various preprocessing models. - We construct a CV-NIZK for QMA in the quantum secret parameter model where a trusted setup sends a quantum proving key to the prover and a classical verification key to the verifier. It is information theoretically sound and zero-knowledge. - Assuming the quantum hardness of the learning with errors problem, we construct a CV-NIZK for QMA in a model where a trusted party generates a CRS and the verifier sends an instance-independent quantum message to the prover as preprocessing. This model is the same as one considered in the recent work by Coladangelo, Vidick, and Zhang (CRYPTO '20). Our construction has the so-called dual-mode property, which means that there are two computationally indistinguishable modes of generating CRS, and we have information theoretical soundness in one mode and information theoretical zero-knowledge property in the other. This answers an open problem left by Coladangelo et al, which is to achieve either of soundness or zero-knowledge information theoretically. To the best of our knowledge, ours is the first dual-mode NIZK for QMA in any kind of model. - We construct a CV-NIZK for QMA with quantum preprocessing in the quantum random oracle model. This quantum preprocessing is the one where the verifier sends a random Pauli-basis states to the prover. Our construction uses the Fiat-Shamir transformation. The quantum preprocessing can be replaced with the setup that distributes Bell pairs among the prover and the verifier, and therefore we solve the open problem by Broadbent and Grilo (FOCS '20) about the possibility of NIZK for QMA in the shared Bell pair model via the Fiat-Shamir transformation.
Peter P. Rohde, Vijay Mohan, Sinclair Davidson, Chris Berg · 7 authors
Two of the most important technological advancements currently underway are the advent of quantum technologies, and the transitioning of global financial systems towards cryptographic assets, notably blockchain-based cryptocurrencies and smart contracts. There is, however, an important interplay between the two, given that, in due course, quantum technology will have the ability to directly compromise the cryptographic foundations of blockchain. We explore this complex interplay by building financial models for quantum failure in various scenarios, including pricing quantum risk premiums. We call this quantum crypto-economics.
Alexandru Cojocaru, Juan A. Garay, Aggelos Kiayias, Fang Song · 5 authors
A proof of work (PoW) is an important cryptographic construct enabling a party to convince others that they invested some effort in solving a computational task. Arguably, its main impact has been in the setting of cryptocurrencies such as Bitcoin and its underlying blockchain protocol, which received significant attention in recent years due to its potential for various applications as well as for solving fundamental distributed computing questions in novel threat models. PoWs enable the linking of blocks in the blockchain data structure and thus the problem of interest is the feasibility of obtaining a sequence (chain) of such proofs. In this work, we examine the hardness of finding such chain of PoWs against quantum strategies. We prove that the chain of PoWs problem reduces to a problem we call multi-solution Bernoulli search, for which we establish its quantum query complexity. Effectively, this is an extension of a threshold direct product theorem to an average-case unstructured search problem. Our proof, adding to active recent efforts, simplifies and generalizes the recording technique of Zhandry (Crypto'19). As an application, we revisit the formal treatment of security of the core of the Bitcoin consensus protocol, the Bitcoin backbone (Eurocrypt'15), against quantum adversaries, while honest parties are classical and show that protocol's security holds under a quantum analogue of the classical “honest majority'' assumption. Our analysis indicates that the security of Bitcoin backbone is guaranteed provided the number of adversarial quantum queries is bounded so that each quantum query is worth <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> classical ones, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math> is the success probability of a single classical query to the protocol's underlying hash function. Somewhat surprisingly, the wait time for safe settlement in the case of quantum adversaries matches the safe settlement time in the classical case.
Pouriya Alikhani, Nicolas Brunner, Claude Crépeau, Sébastien Designolle · 8 authors
Protecting secrets is a key challenge in our contemporary information-based era. In common situations, however, revealing secrets appears unavoidable, for instance, when identifying oneself in a bank to retrieve money. In turn, this may have highly undesirable consequences in the unlikely, yet not unrealistic, case where the bank's security gets compromised. This naturally raises the question of whether disclosing secrets is fundamentally necessary for identifying oneself, or more generally for proving a statement to be correct. Developments in computer science provide an elegant solution via the concept of zero-knowledge proofs: a prover can convince a verifier of the validity of a certain statement without facilitating the elaboration of a proof at all. In this work, we report the experimental realisation of such a zero-knowledge protocol involving two separated verifier-prover pairs. Security is enforced via the physical principle of special relativity, and no computational assumption (such as the existence of one-way functions) is required. Our implementation exclusively relies on off-the-shelf equipment and works at both short (60 m) and long distances ($\geqslant$400 m) in about one second. This demonstrates the practical potential of multi-prover zero-knowledge protocols, promising for identification tasks and blockchain applications such as cryptocurrencies or smart contracts.
Open access
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Cryptography and Data Security
Physical Unclonable Functions (PUFs) and Hardware Security
Prabhanjan Ananth, Kai-Min Chung, Rolando L. La Placa
We study the notion of zero-knowledge secure against quantum polynomial-time verifiers (referred to as quantum zero-knowledge) in the concurrent composition setting. Despite being extensively studied in the classical setting, concurrent composition in the quantum setting has hardly been studied. We initiate a formal study of concurrent quantum zero-knowledge. Our results are as follows: -Bounded Concurrent QZK for NP and QMA: Assuming post-quantum one-way functions, there exists a quantum zero-knowledge proof system for NP in the bounded concurrent setting. In this setting, we fix a priori the number of verifiers that can simultaneously interact with the prover. Under the same assumption, we also show that there exists a quantum zero-knowledge proof system for QMA in the bounded concurrency setting. -Quantum Proofs of Knowledge: Assuming quantum hardness of learning with errors (QLWE), there exists a bounded concurrent zero-knowledge proof system for NP satisfying quantum proof of knowledge property. Our extraction mechanism simultaneously allows for extraction probability to be negligibly close to acceptance probability (extractability) and also ensures that the prover's state after extraction is statistically close to the prover's state after interacting with the verifier (simulatability). The seminal work of [Unruh EUROCRYPT'12], and all its followups, satisfied a weaker version of extractability property and moreover, did not achieve simulatability. Our result yields a proof of quantum knowledge system for QMA with better parameters than prior works.
Michele Ciampi, Alexandru Cojocaru, Elham Kashefi, Atul Mantri
Secure two-party computation considers the problem of two parties computing a\njoint function of their private inputs without revealing anything beyond the\noutput. In this work, we consider the setting where the two parties (a\nclassical Alice and a quantum Bob) can communicate only via a classical\nchannel. Our first result shows that it is in general impossible to realize a\ntwo-party quantum functionality with black-box simulation in the case of\nmalicious quantum adversaries. In particular, we show that the existence of a\nsecure quantum computing protocol that relies only on classical channels would\ncontradict the quantum no-cloning argument.\n We circumvent this impossibility following three different approaches. The\nfirst is by considering a weaker security notion called one-sided simulation\nsecurity. This notion protects the input of one party (the quantum Bob) in the\nstandard simulation-based sense and protects the privacy of the other party's\ninput (the classical Alice). We show how to realize a protocol that satisfies\nthis notion relying on the learning with errors assumption. The second way to\ncircumvent the impossibility result, while at the same time providing standard\nsimulation-based security also against a malicious Bob, is by assuming that the\nquantum input has an efficient classical representation.\n Finally, we focus our attention on the class of zero-knowledge\nfunctionalities and provide a compiler that takes as input a classical proof of\nquantum knowledge (PoQK) protocol for a QMA relation R and outputs a\nzero-knowledge PoQK for R that can be verified by classical parties. The direct\nimplication of our result is that Mahadev's protocol for classical verification\nof quantum computations (FOCS'18) can be turned into a zero-knowledge proof of\nquantum knowledge with classical verifiers. To the best of our knowledge, we\nare the first to instantiate such a primitive.\n
Zainab Abohashima, Mohamed Elhosen, Essam H. Houssein, Waleed M. Mohamed
Due to the superiority and noteworthy progress of Quantum Computing (QC) in a lot of applications such as cryptography, chemistry, Big data, machine learning, optimization, Internet of Things (IoT), Blockchain, communication, and many more. Fully towards to combine classical machine learning (ML) with Quantum Information Processing (QIP) to build a new field in the quantum world is called Quantum Machine Learning (QML) to solve and improve problems that displayed in classical machine learning (e.g. time and energy consumption, kernel estimation). The aim of this paper presents and summarizes a comprehensive survey of the state-of-the-art advances in Quantum Machine Learning (QML). Especially, recent QML classification works. Also, we cover about 30 publications that are published lately in Quantum Machine Learning (QML). we propose a classification scheme in the quantum world and discuss encoding methods for mapping classical data to quantum data. Then, we provide quantum subroutines and some methods of Quantum Computing (QC) in improving performance and speed up of classical Machine Learning (ML). And also some of QML applications in various fields, challenges, and future vision will be presented.
The posthoc verification protocol [J. F. Fitzsimons, M. Hajdu{\v s}ek, and T. Morimae, Physical Review Letters {\bf120}, 040501 (2018)] enables an information-theoretically-sound non-interactive verification of quantum computing, but the message from the prover to the verifier is quantum and the verifier has to do single-qubit measurements. The Mahadev protocol removes these quantum parts, but the soundness becomes the computational one. In this paper, we construct an information-theoretically-sound non-interactive classical verification protocol for quantum computing with a trusted center. The trusted center sends random BB84 states to the prover, and the classical descriptions of these BB84 states to the verifier. The messages from the center to the prover and the verifier are independent of the instance. By slightly modifying our protocol, we also construct a non-interactive statistical zero-knowledge proof system for QMA with the trusted center.
We put forward the idea that classical blockchains and smart contracts are potentially useful primitives not only for classical cryptography, but for quantum cryptography as well. Abstractly, a smart contract is a functionality that allows parties to deposit funds, and release them upon fulfillment of algorithmically checkable conditions, and can thus be employed as a formal tool to enforce monetary incentives. In this work, we give the first example of the use of smart contracts in a quantum setting. We describe a simple hybrid classical-quantum payment system whose main ingredients are a classical blockchain capable of handling stateful smart contracts, and quantum lightning, a strengthening of public-key quantum money introduced by Zhandry (Eurocrypt'19). Our hybrid payment system employs quantum states as banknotes and a classical blockchain to settle disputes and to keep track of the valid serial numbers. It has several desirable properties: it is decentralized, requiring no trust in any single entity; payments are as quick as quantum communication, regardless of the total number of users; when a quantum banknote is damaged or lost, the rightful owner can recover the lost value.
Blockchain and other Distributed Ledger Technologies (DLTs) have evolved significantly in the last years and their use has been suggested for numerous applications due to their ability to provide transparency, redundancy and accountability. In the case of blockchain, such characteristics are provided through public-key cryptography and hash functions. However, the fast progress of quantum computing has opened the possibility of performing attacks based on Grover's and Shor's algorithms in the near future. Such algorithms threaten both public-key cryptography and hash functions, forcing to redesign blockchains to make use of cryptosystems that withstand quantum attacks, thus creating which are known as post-quantum, quantum-proof, quantum-safe or quantum-resistant cryptosystems. For such a purpose, this article first studies current state of the art on post-quantum cryptosystems and how they can be applied to blockchains and DLTs. Moreover, the most relevant post-quantum blockchain systems are studied, as well as their main challenges. Furthermore, extensive comparisons are provided on the characteristics and performance of the most promising post-quantum public-key encryption and digital signature schemes for blockchains. Thus, this article seeks to provide a broad view and useful guidelines on post-quantum blockchain security to future blockchain researchers and developers.