{
  "abstract": "We study the one-way communication complexity of the $f$-Boolean Hidden Partition problem. Here, Alice is given an $n$-bit string and Bob is given $Ω(n)$ disjoint blocks of its indices together with a string of labels. Under the promise that the evaluations of $f$ on these blocks either agree with all of the labels or disagree with all of them, Bob must determine which is the case using a single message from Alice. This problem generalizes the Boolean Hidden Matching and Hidden Hypermatching problems, which capture the special case where $f$ is the parity function. We establish classical and quantum communication bounds for $f$-Boolean Hidden Partition in terms of the sign degree $d$ of $f$, proving a conjecture of Doriguello and Montanaro (TQC 2020). Their prior work gave logarithmic communication upper bounds for classical protocols when $d \\le 1$ and quantum protocols when $d \\le 2$, as well as polynomial lower bounds for certain structured functions with larger sign degree. We show that for every $f$ of sign degree $d \\ge 2$, the randomized classical communication complexity of this problem is $Θ(n^{1-1/d})$ while its quantum communication complexity lies between $Ω(n^{1-2/d})$ and $\\bO{n^{1-1/\\lceil d/2 \\rceil} \\log n}$. This completely characterizes the functions $f$ admitting polynomial quantum advantage as those for which $d \\ge 2$, with a new infinite family of exponential separations given by the case $d = 2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06789",
  "author_total": 4,
  "authors": [
    {
      "name": "Mark Bun"
    },
    {
      "name": "Joao F. Doriguello"
    },
    {
      "name": "John Kallaugher"
    },
    {
      "name": "Nadezhda Voronova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b1sq2t1jgame3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06789",
  "fulltext_url": null,
  "id": "b1sq2t1jgame3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06789",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7220710907",
  "pdf_url": "https://arxiv.org/pdf/2610.06789",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b1sq2t1jgame3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.06789",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06789"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Characterizing Quantum Advantage for Generalizations of the Boolean Hidden Matching Problem",
  "title_text": "Characterizing Quantum Advantage for Generalizations of the Boolean Hidden Matching Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:01Z",
  "url": "https://research.draconitic.com/paper/b1sq2t1jgame3",
  "venue": "arXiv",
  "versions": []
}