{
  "abstract": "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 ($\\sf QSZK$), introduced by Watrous (FOCS 2002). However, restricting the verifier's question to classical public coins collapses the corresponding class to $\\sf BQP$, 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 ${\\sf QIP}_{\\ell\\text{-}{\\rm bit}}(2)$, where $\\ell$ is the length of the prover's response, and establish: 1. A natural complete characterization of ${\\sf QIP}_{\\ell\\text{-}{\\rm bit}}(2)$ by Multi-State Distinguishability. In particular, Quantum State Distinguishability (QSD) is ${\\sf QIP}_{\\rm bit}$-complete. Since QSD is $\\sf QSZK$-hard, our result places ${\\sf QIP}_{\\ell\\text{-}{\\rm bit}}(2)$, for $\\ell\\geq 2$, in a landscape \"just above\" $\\sf QSZK$. 2. Easy regimes for ${\\sf QIP}_{\\ell\\text{-}{\\rm bit}}(2)$ collapsing to $\\sf QSZK$. We prove that QSD$[a,b]$ (and thus ${\\sf QIP}_{\\rm bit}[a,b]$) is in $\\sf QSZK$ when $a(n)-b(n)\\geq 1/O(\\log n)$, and combine this with an answer compression from ${\\sf QIP}_{\\ell\\text{-}{\\rm bit}}[2,c,s]$ to ${\\sf QIP}_{\\rm bit}$ to obtain another easy regime when $2c>(1+2^{\\ell/2})s$. Remarkably, our improved polarization applies to SD and $\\sf SZK$, resolving an open problem in Sahai and Vadhan (JACM, 2003). 3. Quantum public coins also make the interaction useless: ${\\sf qc}\\text{-}{\\sf QAM}[O(\\sqrt{\\log{n}})]$ with constant gap is in $\\sf BQP$, where ${\\sf qc}\\text{-}{\\sf QAM}[\\ell]$ is a subclass of ${\\sf QIP}_{\\ell\\text{-}{\\rm bit}}(2)$ in which the verifier's question is exactly halves of EPR pairs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39495",
  "author_total": 2,
  "authors": [
    {
      "name": "Zihan Hu"
    },
    {
      "name": "Yupan Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bse97639vge4f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39495",
  "fulltext_url": null,
  "id": "bse97639vge4f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39495",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.CR"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7215086839",
  "pdf_url": "https://arxiv.org/pdf/2609.39495",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bse97639vge4f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2609.39495",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39495"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "On quantum interactive proofs with a laconic prover",
  "title_text": "On quantum interactive proofs with a laconic prover",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:01Z",
  "url": "https://research.draconitic.com/paper/bse97639vge4f",
  "venue": "arXiv",
  "versions": []
}