A proof of quantumness is an efficiently verifiable interactive test that an efficient quantum computer can pass, but all efficient classical computers cannot (under some cryptographic assumption). Such protocols play a crucial role in the certification of quantum devices. Existing single-round protocols based solely on a cryptographic hardness assumption (like asking the quantum computer to factor a large number) require large quantum circuits, whereas multi-round ones use smaller circuits but require experimentally challenging mid-circuit measurements. In this work, we construct efficient single-round proofs of quantumness based on existing knowledge assumptions. While knowledge assumptions have not been previously considered in this context, we show that they provide a natural basis for separating classical and quantum computation. Our work also helps in understanding the interplay between black-box/white-box reductions and cryptographic assumptions in the design of proofs of quantumness. Specifically, we show that multi-round protocols based on Decisional Diffie-Hellman (DDH) or Learning With Errors (LWE) can be “compiled” into single-round protocols using a knowledge-of-exponent assumption [7] or knowledge-of-lattice-point assumption [36], respectively. We also prove an adaptive hardcore-bit statement for a family of claw-free functions based on DDH, which might be of independent interest.
A proof of quantumness is an efficiently verifiable interactive test that an efficient quantum computer can pass, but all efficient classical computers cannot (under some cryptographic assumption). Such protocols play a crucial role in the certification of quantum devices. Existing single-round protocols based solely on a cryptographic hardness assumption (like asking the quantum computer to factor a large number) require large quantum circuits, whereas multi-round ones use smaller circuits but require experimentally challenging mid-circuit measurements. In this work, we construct efficient single-round proofs of quantumness based on existing knowledge assumptions. While knowledge assumptions have not been previously considered in this context, we show that they provide a natural basis for separating classical and quantum computation. Our work also helps in understanding the interplay between black-box/white-box reductions and cryptographic assumptions in the design of proofs of quantumness. Specifically, we show that multi-round protocols based on Decisional Diffie-Hellman (DDH) or Learning With Errors (LWE) can be “compiled” into single-round protocols using a knowledge-of-exponent assumption [7] or knowledge-of-lattice-point assumption [36], respectively. We also prove an adaptive hardcore-bit statement for a family of claw-free functions based on DDH, which might be of independent interest.
A proof of quantumness is an efficiently verifiable interactive test that an efficient quantum computer can pass, but all efficient classical computers cannot (under some cryptographic assumption). Such protocols play a crucial role in the certification of quantum devices. Existing single-round protocols (like asking the quantum computer to factor a large number) require large quantum circuits, whereas multi-round ones use smaller circuits but require experimentally challenging mid-circuit measurements. As such, current proofs of quantumness are out of reach for near-term devices. In this work, we construct efficient single-round proofs of quantumness based on existing knowledge assumptions. While knowledge assumptions have not been previously considered in this context, we show that they provide a natural basis for separating classical and quantum computation. Specifically, we show that multi-round protocols based on Decisional Diffie-Hellman (DDH) or Learning With Errors (LWE) can be "compiled" into single-round protocols using a knowledge-of-exponent assumption or knowledge-of-lattice-point assumption, respectively. We also prove an adaptive hardcore-bit statement for a family of claw-free functions based on DDH, which might be of independent interest. Previous approaches to constructing single-round protocols relied on the random oracle model and thus incurred the overhead associated with instantiating the oracle with a cryptographic hash function. In contrast, our protocols have the same resource requirements as their multi-round counterparts without necessitating mid-circuit measurements, making them, arguably, the most efficient single-round proofs of quantumness to date. Our work also helps in understanding the interplay between black-box/white-box reductions and cryptographic assumptions in the design of proofs of quantumness.
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.
An experimental cryptographic proof of quantumness will be a vital milestone in the progress of quantum information science. Error tolerance is a persistent challenge for implementing such tests: we need a test that not only can be passed by an efficient quantum prover, but one that can be passed by a prover that exhibits a certain amount of computational error. (Brakerski et al. 2018) introduced an innovative two-round proof of quantumness based on the Learning With Errors (LWE) assumption. However, one of the steps in their protocol (the pre-image test) has low tolerance for error. In this work we present a proof of quantumness which maintains the same circuit structure as (Brakerski et al. 2018) while improving the robustness for noise. Our protocol is based on cryptographically hiding an extended Greenberger-Horne-Zeilinger (GHZ) state within a sequence of classical bits. Asymptotically, our protocol allows the total probability of error within the circuit to be as high as 1−O(λ−C), where λ is the security parameter and C is a constant that can be made arbitrarily large. As part of the proof of this result, we also prove an uncertainty principle over finite abelian groups which may be of independent interest.
A proof of quantumness is a method for provably demonstrating (to a classical verifier) that a quantum device can perform computational tasks that a classical device with comparable resources cannot. Providing a proof of quantumness is the first step towards constructing a useful quantum computer. There are currently three approaches for exhibiting proofs of quantumness: (i) Inverting a classically-hard one-way function (e.g. using Shor's algorithm). This seems technologically out of reach. (ii) Sampling from a classically-hard-to-sample distribution (e.g. BosonSampling). This may be within reach of near-term experiments, but for all such tasks known verification requires exponential time. (iii) Interactive protocols based on cryptographic assumptions. The use of a trapdoor scheme allows for efficient verification, and implementation seems to require much less resources than (i), yet still more than (ii). In this work we propose a significant simplification to approach (iii) by employing the random oracle heuristic. (We note that we do not apply the Fiat-Shamir paradigm.) We give a two-message (challenge-response) proof of quantumness based on any trapdoor claw-free function. In contrast to earlier proposals we do not need an adaptive hard-core bit property. This allows the use of smaller security parameters and more diverse computational assumptions (such as Ring Learning with Errors), significantly reducing the quantum computational effort required for a successful demonstration.
We give a sampling problem that is solvable by shallow quantum circuits, hard for polynomial-time classical algorithms under lattice-based assumptions, and efficiently verifiable by a classical computer. The quantum sampler admits two implementations: one uses log-logarithmic-depth quantum circuits with one- and two-qubit gates, i.e., QNC0[loglog] circuits, while the other uses constant-depth quantum circuits with unbounded fan-in gates, i.e., QAC0 circuits. Our construction can be seen as compiling the Learning with Errors (LWE)-based single-round proof of quantumness of Arabadjieva et al. (2025) to very low depth. The price paid for this compilation is the reliance on less standard, though well-motivated, assumptions: in addition to the lattice knowledge assumption used by Arabadjieva et al. (2025), we require a strengthened variant of the adaptive-hardcore-bit property of LWE, for which we provide supporting evidence. Unlike previous low-depth proofs of quantumness, the quantum computation here requires no mid-circuit measurements or feed-forward: it consists only of running a shallow circuit and sampling from its output distribution. This shows that shallow quantum circuits have sufficient structure to solve certain classically hard tasks whose solutions can be verified efficiently.