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    "story_id": "mp-2026-10-29-013",
    "source_story_id": "tmp-story-classical-quantum-verification",
    "edition_id": "mp-2026-10-29-morning-0111",
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    "headline": "A Classical Verifier Reaches Beyond the Polynomial Hierarchy",
    "slug": "a-classical-verifier-reaches-beyond-the-polynomial-hierarchy",
    "dek": "Relative to a classical oracle, a public-coin protocol verifies an exactly quantum-decidable language outside the polynomial hierarchy with perfect completeness.",
    "summary": "Relative to a classical oracle, a public-coin protocol verifies an exactly quantum-decidable language outside the polynomial hierarchy with perfect completeness.",
    "body_text": "Whether a classical polynomial-time verifier can efficiently check arbitrary quantum computation without computational assumptions remains open. This paper proves an oracle-relative result for a language outside the polynomial hierarchy but exactly decidable in quantum polynomial time. Its public-coin protocol has perfect completeness and soundness error at most one quarter, using recursively addressed code and data tables. The author also proves a switching theorem for the correlated restrictions that defeat standard tools. Because the construction is relative to a chosen oracle, it does not resolve unrelativized verification of arbitrary real quantum computations.",
    "why_it_matters": "Relative to a classical oracle, a public-coin protocol verifies an exactly quantum-decidable language outside the polynomial hierarchy with perfect completeness.",
    "limitations": [
      "Because the construction is relative to a chosen oracle, it does not resolve unrelativized verification of arbitrary real quantum computations."
    ],
    "importance": 9,
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    "first_published_at": "2026-10-29T09:00:00.000-04:00",
    "modified_at": "2026-10-29T09:00:00.000-04:00",
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        "text": "Relative to a classical oracle, a public-coin protocol verifies an exactly quantum-decidable language outside the polynomial hierarchy with perfect completeness.",
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        "qualification": "Because the construction is relative to a chosen oracle, it does not resolve unrelativized verification of arbitrary real quantum computations."
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    "source_ids": [
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    "tags": [
      "quantum computing",
      "verification",
      "complexity theory",
      "interactive proofs"
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  "sources": [
    {
      "source_id": "source-2026-10-29-013",
      "title": "arXiv preprint 2610.37635, original v1",
      "publisher": "arXiv",
      "url": "https://arxiv.org/abs/2610.37635v1",
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      "source_type": "primary_research",
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      "published_at": "2026-10-26T20:00:00.000-04:00",
      "accessed_at": "2026-10-29T08:20:13.210-04:00",
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    "name": "The Machine Press",
    "url": "https://themachinepress.com",
    "description": "A daily newspaper for the age of artificial intelligence."
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  "cite_this_report": {
    "title": "A Classical Verifier Reaches Beyond the Polynomial Hierarchy",
    "publisher": "The Machine Press",
    "published_at": "2026-10-29T09:00:00.000-04:00",
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