In this paper, we introduce a new concept, proofs of quantum memory (PoQM). A PoQM is an interactive protocol between a classical verifier and a quantum prover over a classical channel where the verifier can verify that the prover has possessed a specified number of qubits for a specified period of time. PoQM generalize proofs of quantumness (PoQ) [Brakerski, Christiano, Mahadev, Vazirani, and Vidick, JACM 2021] where a classical verifier can verify that the prover is not classical. Our contributions are summarized as follows: We introduce PoQM. We also introduce a variant of PoQM, which we call inefficient-verifier PoQM (IV-PoQM), where the verifier's final computation to make the decision is not necessarily efficient. We construct PoQM based on the hardness of LWE. Specifically, we give two constructions of PoQM. The first one is of two-round and has negligible soundness error under the subexponential-hardness of LWE. The second one is of polynomial-round and has inverse-polynomial soundness error under the polynomial-hardness of LWE. We show that IV-PoQM imply one-way puzzles. Moreover, we show that a certain restricted version of PoQM implies quantum computation classical communication key exchange, which suggests the difficulty of black-box constructing PoQM from one-way functions. We show that constant-round PoQ imply PoQM or single-round PoQ (with a quantum verifier). Single-round PoQ are ``trivial'' PoQ in the sense that the verifier asks the prover to solve a classical problem which is quantumly easy but classically hard. The result therefore demonstrates that PoQM capture ``genuinely-interactive'' PoQ. We show that if constant-round IV-PoQ that are black-box constructed from quantumly-secure falsifiable assumptions exist then IV-PoQM exist. This result implies that IV-PoQM can be constructed from quantumly-secure constant-round statistically-hiding commitments.
In this paper, we introduce a new concept, proofs of quantum memory (PoQM). A PoQM is an interactive protocol between a classical verifier and a quantum prover over a classical channel where the verifier can verify that the prover has possessed a specified number of qubits for a specified period of time. PoQM generalize proofs of quantumness (PoQ) [Brakerski, Christiano, Mahadev, Vazirani, and Vidick, JACM 2021] where a classical verifier can verify that the prover is not classical. Our contributions are summarized as follows: We introduce PoQM. We also introduce a variant of PoQM, which we call inefficient-verifier PoQM (IV-PoQM), where the verifier's final computation to make the decision is not necessarily efficient. We construct PoQM based on the hardness of LWE. Specifically, we give two constructions of PoQM. The first one is of two-round and has negligible soundness error under the subexponential-hardness of LWE. The second one is of polynomial-round and has inverse-polynomial soundness error under the polynomial-hardness of LWE. We show that IV-PoQM imply one-way puzzles. Moreover, we show that a certain restricted version of PoQM implies quantum computation classical communication key exchange, which suggests the difficulty of black-box constructing PoQM from one-way functions. We show that constant-round PoQ imply PoQM or single-round PoQ (with a quantum verifier). Single-round PoQ are trivial'' PoQ in the sense that the verifier asks the prover to solve a classical problem which is quantumly easy but classically hard. The result therefore demonstrates that PoQM capture genuinely-interactive'' PoQ. We show that if constant-round IV-PoQ that are black-box constructed from quantumly-secure falsifiable assumptions exist then IV-PoQM exist. This result implies that IV-PoQM can be constructed from quantumly-secure constant-round statistically-hiding commitments.
In this work, we study the hardness required to achieve proofs of quantumness (PoQ), which in turn capture (potentially interactive) quantum advantage. A “trivial” or non-interactive PoQ simply assumes an (efficiently-verifiable) average-case hard problem for classical computers that is easy for quantum computers. However, there is much interest in “non-trivial” PoQs that actually rely on quantum hardness assumptions, instead of an assumed separation between quantum and classical computation for search problems, especially since these are often a starting point for more sophisticated protocols such as classical verification of quantum computation (CVQC). We show several lower-bounds for the hardness required to achieve non-trivial PoQ, specifically showing that they likely require cryptographic hardness, with different types of cryptographic hardness being required for different variations of non-trivial PoQ. In particular, our results help explain the challenges in using lattices to build publicly verifiable PoQ and its various extensions such as CVQC.
With the rapid development of quantum computers, proofs of quantumness have recently become an interesting research direction. However, in current schemes for proofs of quantumness, quantum provers face the risk of being maliciously exploited by classical verifiers. Through malicious strategies in interaction with quantum provers, classical verifiers could solve some instances of hard problems that arise from the specific scheme in use. This is due to the lack of formalization that prevents malicious verifiers from extracting useful information in proofs of quantumness. To address this issue, we formalize zero-knowledge proofs of quantumness. Intuitively, the zero-knowledge property necessitates that the information gained by the classical verifier from interactions with the quantum prover should not surpass what can be simulated using a simulated classical prover interacting with the same verifier. As a result, the new zero-knowledge notion can prevent a malicious verifier from exploiting quantum advantage. We find that the classical zero-knowledge proof is sufficient to compile some existing proofs of quantumness schemes into zero-knowledge proofs of quantumness schemes. It appears to be more general to require zero-knowledge proof on the verifier side instead of the prover side. This helps to regulate the verifier's behavior from malicious to be honest-but-curious. As a result, both parties will play not only one role in the proofs of quantumness but also the dual role in the classical zero-knowledge proof. Specifically, Shor's factoring-based scheme and the learning with errors-based scheme in [Brakerski et al., FOCS, 2018] can be transformed into zero-knowledge proofs of quantumness by requiring an extractable non-interactive zero-knowledge argument on the verifier side. Zero-knowledge proofs of quantumness can thus be viewed as an enhanced security notion for proofs of quantumness.
A proof of quantumness is a type of challenge-response protocol in which a classical verifier can efficiently certify the quantum advantage of an untrusted prover. That is, a quantum prover can correctly answer the verifier's challenges and be accepted, while any polynomial-time classical prover will be rejected with high probability, based on plausible computational assumptions. To answer the verifier's challenges, existing proofs of quantumness typically require the quantum prover to perform a combination of polynomial-size quantum circuits and measurements. In this paper, we give two proof of quantumness constructions in which the prover need only perform constant-depth quantum circuits (and measurements) together with log-depth classical computation. Our first construction is a generic compiler that allows us to translate all existing proofs of quantumness into constant quantum depth versions. Our second construction is based around the learning with rounding problem, and yields circuits with shorter depth and requiring fewer qubits than the generic construction. In addition, the second construction also has some robustness against noise.
We initiate the study of quantum Interactive Oracle Proofs (qIOPs), a generalization of both quantum Probabilistically Checkable Proofs and quantum Interactive Proofs, as well as a quantum analogue of classical Interactive Oracle Proofs.
In the model of quantum Interactive Oracle Proofs, we allow multiple rounds of quantum interaction between the quantum prover and the quantum verifier, but the verifier has limited access to quantum resources. This includes both queries to the prover's messages and the complexity of the quantum circuits applied by the verifier. The question of whether QMA admits a quantum interactive oracle proof system is a relaxation of the quantum PCP Conjecture.
We show the following two main constructions of qIOPs, both of which are unconditional:
- We construct a qIOP for QMA in which the verifier shares polynomially many EPR pairs with the prover at the start of the protocol and reads only a constant number of qubits from the prover's messages.
- We provide a stronger construction of qIOP for QMA in which the verifier not only reads a constant number of qubits but also operates on a constant number of qubits overall, including those in their private registers. However, in this stronger setting, the communication complexity becomes exponential. This leaves open the question of whether strong qIOPs for QMA, with polynomial communication complexity, exist.
As a key component of our construction, we introduce a novel single prover many-qubits test, which may be of independent interest.
Properties of quantum mechanics such as entanglement and the no-cloning theorem enable new forms of cryptography that are unattainable in classical models of computation. Quantum information, however, is harder to process, store, and transmit than classical information. As such, it is desirable to construct semi-quantum cryptographic protocols where only a subset of the parties have quantum capabilities while the rest operate classically. In this dissertation, we analyze the security of semi-quantum protocols based on two approaches: (i) nonlocality-based techniques and (ii) the hardness of the Learning with Errors (LWE) problem. In the first part of the dissertation, we leverage a form of Bell test to construct a quantum position verification (QPV) protocol. Initial QPV protocols required verifiers to prepare and send quantum states to an honest prover to achieve security against adversaries with bounded memory or entanglement. Later work constructed QPV protocols with classical verifiers but required additional assumptions such as the hardness of LWE or of random circuit sampling. We construct a variant of QPV where classical verifiers verify the spatial locations of two provers. We use the nonlocal correlations produced by the two provers to show security against adversaries with bounded quantum memory without making any computational intractability assumptions. In the second part of the dissertation, we use trapdoor claw-free functions (TCFs) to construct semi-quantum cryptographic protocols. TCFs, which are constructible from the hardness of LWE, have been shown to be immensely valuable in the construction of classically verifiable quantum cryptographic protocols. As opposed to previous work where the server is asked to prepare a claw state and perform Pauli-X and Z measurements, we use a technique that leverages the entire range of qubit measurements on the XY-plane. We use this technique to construct three protocols: (i) a one-round blind remote state preparation protocol that delegates the preparation of an arbitrary state on the XY-plane up to a Pauli-X correction; (ii) a two-round proof of quantumness whose security is directly in terms of the hardness of LWE and only requires the prover to prepare and measure a single claw state; and (iii) a 1-of-2 Puzzle with a single qubit auxiliary state.
In a recent seminal work, Bitansky and Shmueli (STOC '20) gave the first construction of a constant round zero-knowledge argument for NP secure against quantum attacks. However, their construction has several drawbacks compared to the classical counterparts. Specifically, their construction only achieves computational soundness, requires strong assumptions of quantum hardness of learning with errors (QLWE assumption) and the existence of quantum fully homomorphic encryption (QFHE), and relies on non-black-box simulation. In this paper, we resolve these issues at the cost of weakening the notion of zero-knowledge to what is called ε-zero-knowledge. Concretely, we construct the following protocols: - We construct a constant round interactive proof for NP that satisfies statistical soundness and black-box ε-zero-knowledge against quantum attacks assuming the existence of collapsing hash functions, which is a quantum counterpart of collision-resistant hash functions. Interestingly, this construction is just an adapted version of the classical protocol by Goldreich and Kahan (JoC '96) though the proof of ε-zero-knowledge property against quantum adversaries requires novel ideas. - We construct a constant round interactive argument for NP that satisfies computational soundness and black-box ε-zero-knowledge against quantum attacks only assuming the existence of post-quantum one-way functions. At the heart of our results is a new quantum rewinding technique that enables a simulator to extract a committed message of a malicious verifier while simulating verifier's internal state in an appropriate sense.