Quantum computational advantage refers to an existence of computational tasks that are easy for quantum computing but hard for classical one. Unconditionally showing quantum advantage is beyond our current understanding of complexity theory, and therefore some computational assumptions are needed. Which complexity assumption is necessary and sufficient for quantum advantage? In this paper, we show that inefficient-verifier proofs of quantumness (IV-PoQ) exist if and only if classically-secure one-way puzzles (OWPuzzs) exist. As far as we know, this is the first time that a complete cryptographic characterization of quantum advantage is obtained. IV-PoQ capture various types of quantum advantage previously studied, such as sampling-based quantum advantage and searching-based one. Previous work [Morimae and Yamakawa, Crypto 2024] showed that IV-PoQ can be constructed from OWFs, but a construction of IV-PoQ from weaker assumptions was left open. Our result solves the open problem. OWPuzzs are one of the most fundamental quantum cryptographic primitives implied by many quantum cryptographic primitives weaker than one-way functions (OWFs). The equivalence between IV-PoQ and classically-secure OWPuzzs therefore highlights that if there is no quantum advantage, then these fundamental primitives do not exist. The equivalence also means that quantum advantage is an example of the applications of OWPuzzs. Except for commitments, no application of OWPuzzs was known before. Our result shows that quantum advantage is another application of OWPuzzs, which solves the open question of [Chung, Goldin, and Gray, Crypto 2024]. Moreover, it is the first quantum-computation-classical-communication (QCCC) application of OWPuzzs.
Quantum computational advantage refers to an existence of computational tasks that are easy for quantum computing but hard for classical one. Unconditionally showing quantum advantage is beyond our current understanding of complexity theory, and therefore some computational assumptions are needed. Which complexity assumption is necessary and sufficient for quantum advantage? In this paper, we show that inefficient-verifier proofs of quantumness (IV-PoQ) exist if and only if classically-secure one-way puzzles (OWPuzzs) exist. As far as we know, this is the first time that a complete cryptographic characterization of quantum advantage is obtained. IV-PoQ capture various types of quantum advantage previously studied, such as sampling-based quantum advantage and searching-based one. Previous work [Morimae and Yamakawa, Crypto 2024] showed that IV-PoQ can be constructed from OWFs, but a construction of IV-PoQ from weaker assumptions was left open. Our result solves the open problem. OWPuzzs are one of the most fundamental quantum cryptographic primitives implied by many quantum cryptographic primitives weaker than one-way functions (OWFs). The equivalence between IV-PoQ and classically-secure OWPuzzs therefore highlights that if there is no quantum advantage, then these fundamental primitives do not exist. The equivalence also means that quantum advantage is an example of the applications of OWPuzzs. Except for commitments, no application of OWPuzzs was known before. Our result shows that quantum advantage is another application of OWPuzzs, which solves the open question of [Chung, Goldin, and Gray, Crypto 2024]. Moreover, it is the first quantum-computation-classical-communication (QCCC) application of OWPuzzs.
In the classical model of computation, it is well established that one-way functions (OWF) are minimal for computational cryptography: They are essential for almost any cryptographic application that cannot be realized with respect to computationally unbounded adversaries. In the quantum setting, however, OWFs appear not to be essential (Kretschmer 2021; Ananth et al., Morimae and Yamakawa 2022), and the question of whether such a minimal primitive exists remains open. We consider EFI pairs - efficiently samplable, statistically far but computationally indistinguishable pairs of (mixed) quantum states. Building on the work of Yan (2022), which shows equivalence between EFI pairs and statistical commitment schemes, we show that EFI pairs are necessary for a large class of quantum-cryptographic applications. Specifically, we construct EFI pairs from minimalistic versions of commitments schemes, oblivious transfer, and general secure multiparty computation, as well as from QCZK proofs from essentially any non-trivial language. We also construct quantum computational zero knowledge (QCZK) proofs for all of QIP from any EFI pair. This suggests that, for much of quantum cryptography, EFI pairs play a similar role to that played by OWFs in the classical setting: they are simple to describe, essential, and also serve as a linchpin for demonstrating equivalence between primitives.
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.
One-way puzzles (OWPuzzs) introduced by Khurana and Tomer [STOC 2024] are a natural quantum analogue of one-way functions (OWFs), and one of the most fundamental primitives in ''Microcrypt'' where OWFs do not exist but quantum cryptography is possible. OWPuzzs are implied by almost all quantum cryptographic primitives, and imply several important applications such as non-interactive commitments and multi-party computations. A significant goal in the field of quantum cryptography is to base OWPuzzs on plausible assumptions that will not imply OWFs. In this paper, we base OWPuzzs on hardness of non-collapsing measurements. To that end, we introduce a new complexity class, SampPDQP, which is a sampling version of the decision class PDQP introduced in [Aaronson, Bouland, Fitzsimons, and Lee, ITCS 2016]. We show that if SampPDQP is hard on average for quantum polynomial time, then OWPuzzs exist. SampPDQP is the class of sampling problems that can be solved by a classical polynomial-time algorithm that can make a single query to a non-collapsing measurement oracle, which is a ''magical'' oracle that can sample measurement results on quantum states without collapsing the states. Such non-collapsing measurements are highly unphysical operations that should be hard to realize in quantum polynomial-time. We also study upperbounds of the hardness of SampPDQP. We introduce a new primitive, distributional collision-resistant puzzles (dCRPuzzs), which are a natural quantum analogue of distributional collision-resistant hashing [Dubrov and Ishai, STOC 2006]. We show that dCRPuzzs imply average-case hardness of SampPDQP (and therefore OWPuzzs as well). We also show that two-message honest-statistically-hiding commitments with classical communication and one-shot signatures [Amos, Georgiou, Kiayias, Zhandry, STOC 2020] imply dCRPuzzs.
Founding Quantum Cryptography on Quantum Advantage, or, Towards Cryptography from #P-Hardness
Recent oracle separations [Kretschmer, TQC'21, Kretschmer et. al., STOC'23] have raised the tantalizing possibility of building quantum cryptography from sources of hardness that persist even if the polynomial hierarchy collapses. We realize this possibility by building quantum bit commitments and secure computation from unrelativized, well-studied mathematical problems that are conjectured to be hard for P#P -- such as approximating the permanents of complex Gaussian matrices, or approximating the output probabilities of random quantum circuits. Indeed, we show that as long as any one of the conjectures underlying sampling-based quantum advantage (e.g., BosonSampling, Random Circuit Sampling, IQP, etc.) is true, quantum cryptography can be based on the extremely mild assumption that P#P⊆(io)BQP/qpoly. We prove that the following hardness assumptions are equivalent. (1) The hardness of approximating the probability assigned to a randomly chosen string in the support of certain efficiently sampleable distributions (upto inverse polynomial multiplicative error).(2) The existence of one-way puzzles, where a quantum sampler outputs a pair of classical strings -- a puzzle and its key -- and where the hardness lies in finding the key corresponding to a random puzzle. These are known to imply quantum bit commitments [Khurana and Tomer, STOC'24]. (3) The existence of state puzzles, or one-way state synthesis, where it is hard to synthesize a secret quantum state given a public classical identifier. These capture the hardness of search problems with quantum inputs (secrets) and classical outputs (challenges). These are the first constructions of quantum cryptographic primitives (one-way puzzles, quantum bit commitments, state puzzles) from concrete, well-founded mathematical assumptions that do not imply the existence of classical cryptography.
Pessiland is a world where NP is hard on average but one-way functions (OWFs) do not exist [Impagliazzo 1995]. Because almost all classical cryptographic primitives imply OWFs [Impagliazzo and Luby 1989], there is almost no classical cryptography in Pessiland. On the other hand, quantum cryptography can exist even when OWFs do not [Kretschmer 2021; Morimae and Yamakawa 2022; Ananth, Qian and Yuen 2022]. Is there a quantum analogue of Pessiland where NP is hard on average but even quantum cryptography does not exist? In this paper, we show that such a miserable world, Quantum Pessiland, exists: there is a quantum oracle relative to which UP∩coUP is hard on average against quantum polynomial-time algorithms with quantum advice, yet auxiliary-input EFI pairs do not exist. We also show that there is a classical oracle relative to which UP∩coUP is hard on average against quantum polynomial-time algorithms with quantum advice, yet classically-secure auxiliary-input one-way puzzles (OWPuzzs) do not exist. Almost all quantum cryptographic primitives imply EFI pairs or OWPuzzs, and therefore these results mean that there is almost no quantum cryptography relative to these oracles. We further show that relative to the classical oracle, SampBQP = SampBPP, and therefore there is no sampling-based quantum advantage in Quantum Pessiland. Finally, because our average-case hardness of UP∩coUP implies P#P⊆i.o.BQP/qpoly, our result also implies that a non-relativizing proof technique is necessary to construct OWPuzzs solely from P#P⊆i.o.BQP/qpoly, which gives a partial negative answer to the open problem of [Khurana and Tomer 2025].
In classical cryptography, one-way functions are necessary for most of cryptography. In the quantum world, however, Kretschmer at {TQC'21} and subsequent works have demonstrated that quantum versions of various primitives in minicrypt, such as signatures and private-key encryption, do not require quantum-computable one-way functions (\textsf{qOWF}s). However, the question of the necessity of \textsf{qOWF}s for primitives in Cryptomania like public-key encryption with classical public keys and ciphertexts (\textsf{qPKE}) has remained open.
We provide a comprehensive picture of the relationship between \textsf{qPKE} and \textsf{qOWF}. First, our main result is a black-box separation between \textsf{qOWF}s and \textsf{qPKE}. In particular, we show that relative to a unitary, real, and self-inverse oracle, there exists a \textsf{qPKE} scheme (with classical keys and ciphertexts), but \textsf{qOWF}s do not exist. Further, our oracles admit a point-function obfuscation construction, thus ruling out black-box construction of \textsf{qOWF} from point function obfuscations.
On the positive side, we construct \textsf{qPKE} from quantum indistinguishability obfuscation for quantum circuits with classical obfuscated descriptions and efficiently verifiable one-way puzzles, which is known to be a potentially weaker assumption than \textsf{qOWF}s, due to an existing black-box separation. Our construction is the quantum analog of the classical construction of \textsf{PKE} from one-way functions and indistinguishability obfuscation in the quantum computation with classical communication (QCCC) setting.