Prover as Search-to-Decision and a Holographic PCP as the Verifier

The model of interactive oracle proofs (IOP) generalizes the notion of probabilistically checkable proof (PCP), in which a static proof is verified probabilistically by querying a small number of bits, to the interactive setting: a polynomial-time verifier interacts with an unbounded prover, but is restricted to only reading a small number of bits, in total, from the messages sent by the prover. IOPs provide a relaxed setting in which to study local probabilistic verification. They have proved instrumental in devising efficient methods for verification through subsequent compilation into non-interactive or succinct protoocls. We study a quantum analogue of interactive oracle proofs (qIOP) in which the verifier and communication are both allowed to be quantum; yet the verifier is restricted to perform measurements only on a small number of qubits received from the prover. Our main result is a qIOP for any language in QMA, in which the total communication is polynomial but the verifier only reads a polylogarithmic number of qubits in total. The protocol has completeness parameter exponentially close to 1 and soundness bounded away from 1 by a constant. In the absence of a quantum PCP theorem, this provides the first information-theoretically sound local and robust characterization of QMA, albeit interactive. Previous works in the information-theoretic setting either considered two isolated but entangled quantum provers or quantum verifiers whose effort in a single round is small but remains polynomial when aggregated across all rounds of the protocol. Our protocol combines the use of a quantum locally testable code (LTC) with classical techniques, notably probabilistically checkable proofs of proximity (PCPP). We avoid the necessity for complex multi-qubit tests employed in other settings by leveraging the local indistinguishability property of the quantum LTC.The model of interactive oracle proofs (IOP) generalizes the notion of probabilistically checkable proof (PCP), in which a static proof is verified probabilistically by querying a small number of bits, to the interactive setting: a polynomial-time verifier interacts with an unbounded prover, but is restricted to only reading a small number of bits, in total, from the messages sent by the prover. IOPs provide a relaxed setting in which to study local probabilistic verification. They have proved instrumental in devising efficient methods for verification through subsequent compilation into non-interactive or succinct protoocls. We study a quantum analogue of interactive oracle proofs (qIOP) in which the verifier and communication are both allowed to be quantum; yet the verifier is restricted to perform measurements only on a small number of qubits received from the prover. Our main result is a qIOP for any language in QMA, in which the total communication is polynomial but the verifier only reads a polylogarithmic number of qubits in total. The protocol has completeness parameter exponentially close to 1 and soundness bounded away from 1 by a constant. In the absence of a quantum PCP theorem, this provides the first information-theoretically sound local and robust characterization of QMA, albeit interactive. Previous works in the information-theoretic setting either considered two isolated but entangled quantum provers or quantum verifiers whose effort in a single round is small but remains polynomial when aggregated across all rounds of the protocol. Our protocol combines the use of a quantum locally testable code (LTC) with classical techniques, notably probabilistically checkable proofs of proximity (PCPP). We avoid the necessity for complex multi-qubit tests employed in other settings by leveraging the local indistinguishability property of the quantum LTC.

Leibniz international proceedings in informaticsJan 1, 20260 citations

Probabilistically Checking Quantum Proofs, with Interaction

Baocheng Sun, Thomas Vidick

The model of interactive oracle proofs (IOP) generalizes the notion of probabilistically checkable proof (PCP), in which a static proof is verified probabilistically by querying a small number of bits, to the interactive setting: a polynomial-time verifier interacts with an unbounded prover, but is restricted to only reading a small number of bits, in total, from the messages sent by the prover. IOPs provide a relaxed setting in which to study local probabilistic verification. They have proved instrumental in devising efficient methods for verification through subsequent compilation into non-interactive or succinct protocols. We study a quantum analogue of interactive oracle proofs (qIOP) in which the verifier and communication are both allowed to be quantum; yet the verifier is restricted to perform measurements only on a small number of qubits received from the prover. Our main result is a qIOP for any language in QMA, in which the total communication is polynomial but the verifier only reads a polylogarithmic number of qubits in total. The protocol has completeness parameter exponentially close to $1$ and soundness bounded away from $1$ by a constant. In the absence of a quantum PCP theorem, this provides the first information-theoretically sound local and robust characterization of QMA, albeit interactive. Our protocol combines the use of a quantum locally testable code (LTC) with classical techniques, notably probabilistically checkable proofs of proximity (PCPP). We avoid the necessity for complex multi-qubit tests employed in other settings by leveraging the local indistinguishability property of the quantum LTC.The model of interactive oracle proofs (IOP) generalizes the notion of probabilistically checkable proof (PCP), in which a static proof is verified probabilistically by querying a small number of bits, to the interactive setting: a polynomial-time verifier interacts with an unbounded prover, but is restricted to only reading a small number of bits, in total, from the messages sent by the prover. IOPs provide a relaxed setting in which to study local probabilistic verification. They have proved instrumental in devising efficient methods for verification through subsequent compilation into non-interactive or succinct protocols. We study a quantum analogue of interactive oracle proofs (qIOP) in which the verifier and communication are both allowed to be quantum; yet the verifier is restricted to perform measurements only on a small number of qubits received from the prover. Our main result is a qIOP for any language in QMA, in which the total communication is polynomial but the verifier only reads a polylogarithmic number of qubits in total. The protocol has completeness parameter exponentially close to $1$ and soundness bounded away from $1$ by a constant. In the absence of a quantum PCP theorem, this provides the first information-theoretically sound local and robust characterization of QMA, albeit interactive. Our protocol combines the use of a quantum locally testable code (LTC) with classical techniques, notably probabilistically checkable proofs of proximity (PCPP). We avoid the necessity for complex multi-qubit tests employed in other settings by leveraging the local indistinguishability property of the quantum LTC.

cs.CCCross-listedarXiv:2606.09588Jun 9, 20260 citations

Some Observations on Holographic Algorithms

Leslie Gabriel Valiant

Lecture notes in computer scienceJan 1, 201015 citations

Multi-Prover Interactive Proof Systems with Leakage

Vahid R. Asadi, Atsuya Hasegawa, François Le Gall

It is known that there exist multi-prover interactive protocols ( $\mathsf{MIP}$ protocols) for the complexity class $\mathsf{NEXP}$ , succinct $\mathsf{MIP}$ protocols for $\mathsf{NP}$ and multi-prover interactive protocols with shared entanglement ( $\mathsf{MIP}^\ast$ protocols) for $\mathsf{RE}$ . This extraordinary power of multi-prover interactive proof systems comes from the assumption that provers do not communicate with each other during the protocols. If they are allowed to communicate freely, the setting is the same as in the single-prover case, and the computational power of the system becomes significantly weaker. In this paper, we investigate for the first time the setting where communication (i.e., leakage of information) between provers is allowed but bounded. We introduce two techniques to approach this question and show that multi-prover interactive proof systems are robust against some amount of leakage. Our first technique is based on parallel repetition theorems. We apply it to show that for any polynomial $p$ , we can construct two-prover one-round $\mathsf{MIP}$ and $\mathsf{MIP}^\ast$ protocols for $\mathsf{NEXP}$ and $\mathsf{RE}$ , respectively, that are robust against $p(n)$ bits of leakage. We further derive our second technique to convert any low-soundness PCP construction to a two-prover one-round $\mathsf{MIP}$ protocol for $\mathsf{NP}$ robust against leakage. We also discuss the relation between robustness against leakage in multi-prover interactive proof systems and the Sliding Scale Conjecture in the PCP literature.It is known that there exist multi-prover interactive protocols ($\mathsf{MIP}$ protocols) for the complexity class $\mathsf{NEXP}$, succinct $\mathsf{MIP}$ protocols for $\mathsf{NP}$ and multi-prover interactive protocols with shared entanglement ($\mathsf{MIP}^\ast$ protocols) for $\mathsf{RE}$. This extraordinary power of multi-prover interactive proof systems comes from the assumption that provers do not communicate with each other during the protocols. If they are allowed to communicate freely, the setting is the same as in the single-prover case, and the computational power of the system becomes significantly weaker. In this paper, we investigate for the first time the setting where communication (i.e., leakage of information) between provers is allowed but bounded. We introduce two techniques to approach this question and show that multi-prover interactive proof systems are robust against some amount of leakage. Our first technique is based on parallel repetition theorems. We apply it to show that for any polynomial $p$, we can construct two-prover one-round $\mathsf{MIP}$ and $\mathsf{MIP}^\ast$ protocols for $\mathsf{NEXP}$ and $\mathsf{RE}$, respectively, that are robust against $p(n)$ bits of leakage. We further derive our second technique to convert any low-soundness PCP construction to a two-prover one-round $\mathsf{MIP}$ protocol for $\mathsf{NP}$ robust against leakage. We also discuss the relation between robustness against leakage in multi-prover interactive proof systems and the Sliding Scale Conjecture in the PCP literature.

quant-pharXiv:2605.09872Leibniz international proceedings in informaticsMay 12, 20260 citations

Some observations on holographic algorithms

Leslie Gabriel Valiant

Computational ComplexityAug 29, 20175 citations

Quantum State Search with Classical Interactive Advice

Mohammed Barhoush

We study whether a quantum verifier can find a hidden quantum state by exchanging classical messages with an unbounded prover who knows the state. The verifier has access to a reflection oracle about the target. For a Haar-random $n$ -qubit target, we prove that achieving constant average fidelity with the target requires $(q+1)^2(B+1)=Ω(2^n)$ , where $q$ is the number of verifier queries and $B$ is the total prover-to-verifier communication. The bound allows arbitrary classical interaction and unrestricted computation between queries. Our proof extends the lower bound of Aaronson and Kuperberg (ToC'07) from a single classical proof to an arbitrary classical interaction. As a result, we obtain a quantum-oracle separation of QMA from QCIP. In particular, we give a unitary oracle $\mathcal{U}$ such that there exists a language $L\in\mathrm{QMA}^{\mathcal{U}}$ but $L\notin\mathrm{QCIP}^{\mathcal{U}}$ . Our separation holds even for UniqueQMA, the restriction of QMA to verification with a unique quantum witness, which is known to be strictly weaker than QMA in the quantum black-box setting. We further give an oracle construction for a notion known as QMA--QCIP[2] gap problem. This gives average case hardness for a QMA language under classical access to a QCIP[2] decision oracle and is used in establishing negative results on witness encryption and quantum indistinguishability obfuscation.We study whether a quantum verifier can find a hidden quantum state by exchanging classical messages with an unbounded prover who knows the state. The verifier has access to a reflection oracle about the target. For a Haar-random $n$-qubit target, we prove that achieving constant average fidelity with the target requires $(q+1)^2(B+1)=Ω(2^n)$, where $q$ is the number of verifier queries and $B$ is the total prover-to-verifier communication. The bound allows arbitrary classical interaction and unrestricted computation between queries. Our proof extends the lower bound of Aaronson and Kuperberg (ToC'07) from a single classical proof to an arbitrary classical interaction. As a result, we obtain a quantum-oracle separation of QMA from QCIP. In particular, we give a unitary oracle $\mathcal{U}$ such that there exists a language $L\in\mathrm{QMA}^{\mathcal{U}}$ but $L\notin\mathrm{QCIP}^{\mathcal{U}}$. Our separation holds even for UniqueQMA, the restriction of QMA to verification with a unique quantum witness, which is known to be strictly weaker than QMA in the quantum black-box setting. We further give an oracle construction for a notion known as QMA--QCIP[2] gap problem. This gives average case hardness for a QMA language under classical access to a QCIP[2] decision oracle and is used in establishing negative results on witness encryption and quantum indistinguishability obfuscation.

quant-pharXiv:2610.25657Oct 20, 20260 citations

1

and soundness bounded away from

1

by a constant. In the absence of a quantum PCP theorem, this provides the first information-theoretically sound local and robust characterization of QMA, albeit interactive. Our protocol combines the use of a quantum locally testable code (LTC) with classical techniques, notably probabilistically checkable proofs of proximity (PCPP). We avoid the necessity for complex multi-qubit tests employed in other settings by leveraging the local indistinguishability property of the quantum LTC.

It is known that there exist multi-prover interactive protocols (

\mathsf{MIP}

protocols) for the complexity class

\mathsf{NEXP}

, succinct

\mathsf{MIP}

protocols for

\mathsf{NP}

and multi-prover interactive protocols with shared entanglement (

\mathsf{MIP}^\ast

protocols) for

\mathsf{RE}

. This extraordinary power of multi-prover interactive proof systems comes from the assumption that provers do not communicate with each other during the protocols. If they are allowed to communicate freely, the setting is the same as in the single-prover case, and the computational power of the system becomes significantly weaker. In this paper, we investigate for the first time the setting where communication (i.e., leakage of information) between provers is allowed but bounded. We introduce two techniques to approach this question and show that multi-prover interactive proof systems are robust against some amount of leakage. Our first technique is based on parallel repetition theorems. We apply it to show that for any polynomial

p

, we can construct two-prover one-round

\mathsf{MIP}

and

\mathsf{MIP}^\ast

protocols for

\mathsf{NEXP}

and

\mathsf{RE}

, respectively, that are robust against

p(n)

bits of leakage. We further derive our second technique to convert any low-soundness PCP construction to a two-prover one-round

\mathsf{MIP}

protocol for

\mathsf{NP}

robust against leakage. We also discuss the relation between robustness against leakage in multi-prover interactive proof systems and the Sliding Scale Conjecture in the PCP literature.

n

-qubit target, we prove that achieving constant average fidelity with the target requires

(q+1)^2(B+1)=Ω(2^n)

, where

q

is the number of verifier queries and

B

is the total prover-to-verifier communication. The bound allows arbitrary classical interaction and unrestricted computation between queries. Our proof extends the lower bound of Aaronson and Kuperberg (ToC'07) from a single classical proof to an arbitrary classical interaction. As a result, we obtain a quantum-oracle separation of QMA from QCIP. In particular, we give a unitary oracle

\mathcal{U}

such that there exists a language

L\in\mathrm{QMA}^{\mathcal{U}}

but

L\notin\mathrm{QCIP}^{\mathcal{U}}

. Our separation holds even for UniqueQMA, the restriction of QMA to verification with a unique quantum witness, which is known to be strictly weaker than QMA in the quantum black-box setting. We further give an oracle construction for a notion known as QMA--QCIP[2] gap problem. This gives average case hardness for a QMA language under classical access to a QCIP[2] decision oracle and is used in establishing negative results on witness encryption and quantum indistinguishability obfuscation.