We introduce two models of space-bounded quantum interactive proof systems, QIPL and QIP_{U}L. The QIP_{U}L model, a space-bounded variant of quantum interactive proofs (QIP) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, QIPL allows logarithmically many pinching intermediate measurements per verifier action, making it the weakest model that encompasses the classical model of Condon and Ladner (JCSS 1995). We characterize the computational power of QIPL and QIP_{U}L. When the message number m is polynomially bounded, QIP_{U}L ⊊ QIPL unless P = NP: - QIPL^HC, a subclass of QIPL defined by a high-concentration condition on yes instances, exactly characterizes NP. - QIP_{U}L is contained in P and contains SAC¹ ∪ BQL, where SAC¹ denotes problems solvable by classical logarithmic-depth, semi-unbounded fan-in circuits. However, this distinction vanishes when m is constant. Our results further indicate that (pinching) intermediate measurements uniquely impact space-bounded quantum interactive proofs, unlike in space-bounded quantum computation, where BQL = BQ_{U}L. We also introduce space-bounded unitary quantum statistical zero-knowledge (QSZK_{U}L), a specific form of QIP_{U}L proof systems with statistical zero-knowledge against any verifier. This class is a space-bounded variant of quantum statistical zero-knowledge (QSZK) defined by Watrous (SICOMP 2009). We prove that QSZK_{U}L = BQL, implying that the statistical zero-knowledge property negates the computational advantage typically gained from the interaction.
We introduce two models of space-bounded quantum interactive proof systems, QIPL and QIP_{U}L. The QIP_{U}L model, a space-bounded variant of quantum interactive proofs (QIP) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, QIPL allows logarithmically many pinching intermediate measurements per verifier action, making it the weakest model that encompasses the classical model of Condon and Ladner (JCSS 1995). We characterize the computational power of QIPL and QIP_{U}L. When the message number m is polynomially bounded, QIP_{U}L ⊊ QIPL unless P = NP: - QIPL^HC, a subclass of QIPL defined by a high-concentration condition on yes instances, exactly characterizes NP. - QIP_{U}L is contained in P and contains SAC¹ ∪ BQL, where SAC¹ denotes problems solvable by classical logarithmic-depth, semi-unbounded fan-in circuits. However, this distinction vanishes when m is constant. Our results further indicate that (pinching) intermediate measurements uniquely impact space-bounded quantum interactive proofs, unlike in space-bounded quantum computation, where BQL = BQ_{U}L. We also introduce space-bounded unitary quantum statistical zero-knowledge (QSZK_{U}L), a specific form of QIP_{U}L proof systems with statistical zero-knowledge against any verifier. This class is a space-bounded variant of quantum statistical zero-knowledge (QSZK) defined by Watrous (SICOMP 2009). We prove that QSZK_{U}L = BQL, implying that the statistical zero-knowledge property negates the computational advantage typically gained from the interaction.
Interactive proof systems with a laconic prover, studied by Goldreich, Vadhan, and Wigderson (CC, 2002), capture problems verifiable with logarithmic prover communication in the classical setting. For two-message quantum analogs, even a single-bit prover response contains quantum statistical zero-knowledge (QSZK), introduced by Watrous (FOCS 2002). However, restricting the verifier's question to classical public coins collapses the corresponding class to BQP, as shown by Beigi, Shor, and Watrous (ToC, 2011).
We further study two-message quantum interactive proof systems with a laconic prover. To this end, we introduce the class QIPℓ-bit(2), where ℓ is the length of the prover's response, and establish:
1. A natural complete characterization of QIPℓ-bit(2) by Multi-State Distinguishability. In particular, Quantum State Distinguishability (QSD) is QIPbit-complete. Since QSD is QSZK-hard, our result places QIPℓ-bit(2), for ℓ≥2, in a landscape "just above" QSZK.
2. Easy regimes for QIPℓ-bit(2) collapsing to QSZK. We prove that QSD[a,b] (and thus QIPbit[a,b]) is in QSZK when a(n)−b(n)≥1/O(logn), and combine this with an answer compression from QIPℓ-bit[2,c,s] to QIPbit to obtain another easy regime when 2c>(1+2ℓ/2)s. Remarkably, our improved polarization applies to SD and SZK, resolving an open problem in Sahai and Vadhan (JACM, 2003).
3. Quantum public coins also make the interaction useless: qc-QAM[O(logn)] with constant gap is in BQP, where qc-QAM[ℓ] is a subclass of QIPℓ-bit(2) in which the verifier's question is exactly halves of EPR pairs.
Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class NP, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class QSZK, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.
The complexity class Quantum Statistical Zero-Knowledge (QSZK), introduced by Watrous (FOCS 2002) and later refined in Watrous (SICOMP, 2009), has the best known upper bound QIP(2)∩co-QIP(2), which was simplified following the inclusion QIP(2)⊆PSPACE established in Jain, Upadhyay, and Watrous (FOCS 2009). Here, QIP(2) denotes the class of promise problems that admit two-message quantum interactive proof systems in which the honest prover is typically computationally unbounded, and co-QIP(2) denotes the complement of QIP(2).
We slightly improve this upper bound to QIP(2)∩co-QIP(2) with a quantum linear-space honest prover. Specifically, the honest prover uses space linear in the size of the transcript of the original QSZK proof system. A similar improvement also applies to the upper bound for the non-interactive variant NIQSZK. Our main techniques are algorithmic versions of the Holevo-Helstrom measurement and the Uhlmann transform, both implementable in quantum linear space, implying polynomial-time complexity in the state dimension, using the recent space-efficient quantum singular value transformation of Le Gall, Liu, and Wang (CC, to appear).
Quantum Proofs provides an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, it discusses non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class QSZK, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*. Quantum Proofs is mainly intended for non-specialists having a basic background in complexity theory and quantum information. A typical reader may be a student or researcher in either area desiring to learn about the fundamentals of the (actively developing) theory of quantum interactive proofs.
The round complexity of interactive proof systems is a key question of practical and theoretical relevance in complexity theory and cryptography. Moreover, results such as QIP = QIP(3) (STOC'00) show that quantum resources significantly help in such a task. In this work, we initiate the study of round compression of protocols in the bounded quantum storage model (BQSM). In this model, the malicious parties have a bounded quantum memory and they cannot store the all the qubits that are transmitted in the protocol. Our main results in this setting are the following: 1. There is a non-interactive (statistical) witness indistinguishable proof for any language in NP (and even QMA) in BQSM in the plain model. We notice that in this protocol, only the memory of the verifier is bounded. 2. Any classical proof system can be compressed in a two-message quantum proof system in BQSM. Moreover, if the original proof system is zero-knowledge, the quantum protocol is zero-knowledge too. In this result, we assume that the prover has bounded memory. Finally, we give evidence towards the “tightness” of our results. First, we show that NIZK in the plain model against BQS adversaries is unlikely with standard techniques. Second, we prove that without the BQS model there is no 2–message zero-knowledge quantum interactive proof, even under computational assumptions.
While quantum interactive proof systems using at least three messages can achieve perfect completeness, as shown by Kitaev and Watrous (STOC 2000), whether perfect completeness is achievable for one- and two-message quantum proof systems has remained open. For the one-message case, whether QMA can achieve perfect completeness was posed as an open problem in Watrous (FOCS 2000) and Aharonov and Naveh (2002); for the two-message case, the corresponding problems were (implicitly) posed in Jain, Upadhyay, and Watrous~(FOCS 2009) and Kobayashi, Le Gall, and Nishimura (SICOMP, 2019).
In this work, we establish that QIP(2), qq-QAM, QAM, and QMA can achieve perfect completeness. Here qq-QAM denotes the class of promise problems admitting two-message quantum-public-coin quantum interactive proof systems in which the verifier's only message consists of half-EPR pairs. Our main technical contributions are the follows:
1. For QMA (and directly for QAM), an exactly constructible block-encoded matrix whose kernel certifies yes instances, constructed from the acceptance operator induced by the verification circuit. This also yields an alternative proof of QCMA=QCMA1.
2. For QIP(2) (and implicitly qq-QAM), a new turn-halving transformation that preserves completeness and ensures that the resulting proof system retains at least two messages, provided that the terminal state before the final measurement is efficiently preparable.