In this paper we consider a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NP-type proof. Specifically, we consider quantum proofs for properties of black-box groups, which are finite groups whose elements are encoded as strings of a given length and whose group operations are performed by a group oracle. We prove that for an arbitrary group oracle there exist succinct (polynomial-length) quantum proofs for the Group Non-Membership problem that can be checked with small error in polynomial time on a quantum computer. Classically this is impossible---it is proved that there exists a group oracle relative to which this problem does not have succinct proofs that can be checked classically with bounded error in polynomial time (i.e., the problem is not in MA relative to the group oracle constructed). By considering a certain subproblem of the Group Non-Membership problem we obtain a simple proof that there exists an oracle relative to which BQP is not contained in MA. Finally, we show that quantum proofs for non-membership and classical proofs for various other group properties can be combined to yield succinct quantum proofs for other group properties not having succinct proofs in the classical setting, such as verifying that a number divides the order of a group and verifying that a group is not a simple group.

One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet in some scenarios there are ways of conveying information with much fewer, even exponentially fewer, qubits than possible classically [1], [2], [3]. Moreover, some of these methods have a very simple structure|they involve only few message exchanges between the communicating parties. We consider the question as to whether every classical protocol may be transformed to a \simpler" quantum protocol|one that has similar eciency, but uses fewer message exchanges.

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We consider a variation of the communication complexity scenario, where the parties are supplied with an extra resource: particles in an entangled quantum state. We note that "quantum nonlocality" can be naturally expressed in the language of communication complexity. These are communication complexity problems where the "output" is embodied in the correlations between the outputs of the individual parties. Without entanglement, the parties must communicate to produce the required correlations; whereas, with entanglement, no communication is necessary to produce the correlations. In this sense, nonlocality proofs can also be viewed as communication complexity problems where the presence of quantum entanglement reduces the amount of necessary communication. We show how to transform examples of nonlocality into more traditional communication complexity problems, where the output is explicitly determined by each individual party. The resulting problems require communication with or without entanglement, but the required communication is less when entanglement is available. All these results are a noteworthy contrast to the well-known fact that entanglement cannot be used to actually simulate or compress classical communication between remote parties.

This paper proves that several interactive proof systems are zeroknowledge against general quantum attacks. This includes the well-known Goldreich-Micali-Wigderson classical zero-knowledge protocols for Graph Isomorphism and Graph 3-Coloring (assuming the existence of quantum computationally concealing commitment schemes in the second case). Also included is a quantum interactive protocol for a complete problem for the complexity class of problems having “honest verifier” quantum statistical zero-knowledge proofs, which therefore establishes that honest verifier and general quantum statistical zero-knowledge are equal: QSZK = QSZK HV. Previously no non-trivial proof systems were known to be zero-knowledge against quantum attacks, except in restricted settings such as the honest-verifier and common reference string models. This paper therefore establishes for the first time that true zero-knowledge is indeed possible in the presence of quantum information and computation.

In this paper we propose a definition for honest verifier quantum statistical zero-knowledge interactive proof systems and study the resulting complexity class, which we denote QSZK

Abstract This paper studies quantum Arthur-Merlin games, whichare a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coin-flips. The following results are proved. ffl For one-message quantum Arthur-Merlin games, whichcorrespond to the complexity class QMA, complete-ness and soundness errors can be reduced exponentially without increasing the length of Merlin's message. Pre-vious constructions for reducing error required a polynomial increase in the length of Merlin's message. Ap-plications of this fact include a proof that logarithmic length quantum certificates yield no increase in powerover BQP and a simple proof that QMA ` PP. ffl In the case of three or more messages, quantum Arthur-Merlin games are equivalent in power to ordinary quantum interactive proof systems. In fact, for any languagehaving a quantum interactive proof system there exists a three-message quantum Arthur-Merlin game in whichArthur's only message consists of just a single coin-flip that achieves perfect completeness and soundness errorexponentially close to 1/2. ffl Any language having a two-message quantum Arthur-Merlin game is contained in BP \Delta PP. This gives somesuggestion that three messages are stronger than two in

Quantum Merlin-Arthur proof systems are a weak form of quantum interactive proof systems, where mighty Merlin as a prover presents a proof in a pure quantum state and Arthur as a verifier performs polynomial-time quantum computation to verify its correctness with high success probability. For a more general treatment, this paper considers quantum “multiple-Merlin”-Arthur proof systems in which Arthur uses multiple quantum proofs unentangled each other for his verification. Although classical multi-proof systems are easily shown to be essentially equivalent to classical single-proof systems, it is unclear whether quantum multi-proof systems collapse to quantum single-proof systems. This paper investigates the possibility that quantum multi-proof systems collapse to quantum single-proof systems, and shows that (i) a necessary and sufficient condition under which the number of quantum proofs is reducible to two and (ii) using multiple quantum proofs does not increase the power of quantum Merlin-Arthur proof systems in the case of perfect soundness. Our proof for the latter result also gives a new characterization of the class NQP, which bridges two existing concepts of “quantum nondeterminism”. It is also shown that (iii) there is a relativized world in which co-NP (actually co-UP) does not have quantum Merlin-Arthur proof systems even with multiple quantum proofs. 1 1

We show how to encode 2n (classical) bits a1,...,a2 n by a single quantum state |Ψ 〉 of size O(n) qubits, such that: for any constant k and any i1,...,ik ∈ {1,...,2 n}, the values of the bits ai1,...,aik can be retrieved from |Ψ 〉 by a one-round Arthur-Merlin interactive protocol of size polynomial in n. This shows how to go around Holevo-Nayak’s Theorem, using Arthur-Merlin proofs. We use the new representation to prove the following results: 1. Interactive proofs with quantum advice: We show that the class QIP/qpoly contains all languages. That is, for any language L (even non-recursive), the membership x ∈ L (for x of length n) can be proved by a polynomial-size quantum interactive proof, where the verifier is a polynomialsize quantum circuit with working space initiated with some quantum state |ΨL,n〉 (depending only on L and n). Moreover, the interactive proof that we give is of only one round, and the messages communicated are classical. 2. PCP with only one query: We show that the membership x ∈ SAT (for x of length n) can be proved by a logarithmic-size quantum state |Ψ〉, together with a polynomial-size classical proof consisting of blocks of length polylog(n) bits each, such that after measuring the state |Ψ 〉 the verifier only needs to read one block of the classical proof. While the first result is a straight forward consequence of the new representation, the second requires an additional machinery of quantum low-degree-test that may be interesting in its own right.

Abstract. We study nondeterministic quantum algorithms for Boolean functions f. Such algorithms have positive acceptance probability on input x iff f(x) = 1. In the setting of query complexity, we show that the nondeterministic quantum complexity of a Boolean function is equal to its “nondeterministic polynomial ” degree. We also prove a quantum-vs.-classical gap of 1 vs. n for nondeterministic query complexity for a total function. In the setting of communication complexity, we show that the nondeterministic quantum complexity of a two-party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix. This implies that the quantum communication complexities of the equality and disjointness functions are n + 1 if we do not allow any error probability. We also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart.