@misc{Moore1999,
  author        = {Moore, Cristopher},
  title         = {Quantum Circuits: Fanout, Parity, and Counting},
  year          = {1999},
  howpublished  = {arXiv:quant-ph/9903046},
  eprint        = {quant-ph/9903046},
  archivePrefix = {arXiv},
  primaryClass  = {quant-ph},
  doi           = {10.48550/arXiv.quant-ph/9903046},
  url           = {https://arxiv.org/abs/quant-ph/9903046v3},
  note          = {Version 3, 17 March 1999. Also ECCC Report TR99-032}
}

@article{GreenHomerMoorePollett2002,
  author        = {Green, Frederic and Homer, Steven and Moore, Cristopher and Pollett, Christopher},
  title         = {Counting, fanout and the complexity of quantum {ACC}},
  journal       = {Quantum Information and Computation},
  volume        = {2},
  number        = {1},
  pages         = {35--65},
  year          = {2002},
  month         = jan,
  publisher     = {Rinton Press},
  doi           = {10.26421/QIC2.1-3},
  url           = {https://www.rintonpress.com/journals/doi/QIC2.1-3.html},
  eprint        = {quant-ph/0106017},
  archivePrefix = {arXiv},
  primaryClass  = {quant-ph},
  note          = {Preprint: arXiv:quant-ph/0106017v1, 4 June 2001}
}

@inproceedings{Rosenthal2021,
  author        = {Rosenthal, Gregory},
  title         = {Bounds on the {$\mathsf{QAC}^{0}$} Complexity of Approximating Parity},
  booktitle     = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  editor        = {Lee, James R.},
  series        = {Leibniz International Proceedings in Informatics (LIPIcs)},
  volume        = {185},
  pages         = {32:1--32:20},
  year          = {2021},
  publisher     = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address       = {Dagstuhl, Germany},
  isbn          = {978-3-95977-177-1},
  issn          = {1868-8969},
  doi           = {10.4230/LIPIcs.ITCS.2021.32},
  url           = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.32}
}

@misc{AnshuDongOuYao2025,
  author        = {Anshu, Anurag and Dong, Yangjing and Ou, Fengning and Yao, Penghui},
  title         = {On the Computational Power of {$\mathsf{QAC}^{0}$} with Barely Superlinear Ancillae},
  year          = {2025},
  howpublished  = {arXiv:2410.06499},
  eprint        = {2410.06499},
  archivePrefix = {arXiv},
  primaryClass  = {quant-ph},
  url           = {https://arxiv.org/abs/2410.06499v4},
  note          = {Version 4, submitted 21 December 2025; PDF dated 23 December 2025}
}

@misc{DongOuYao2025,
  author        = {Dong, Yangjing and Ou, Fengning and Yao, Penghui},
  title         = {Linear-Size {$\mathsf{QAC}^{0}$} Channels: Learning, Testing and Hardness},
  year          = {2025},
  howpublished  = {arXiv:2510.00593},
  eprint        = {2510.00593},
  archivePrefix = {arXiv},
  primaryClass  = {quant-ph},
  url           = {https://arxiv.org/abs/2510.00593v2},
  note          = {Version 2, 8 November 2025}
}

@inproceedings{JoshiTalVasconcelosWright2025,
  author = {Joshi, Malvika Raj and Tal, Avishay and Vasconcelos, Francisca and Wright, John},
  title = {Improved Lower Bounds for {QAC0}},
  booktitle = {Proceedings of the 58th Annual ACM Symposium on Theory of Computing},
  year = {2026},
  pages = {2199--2209},
  doi = {10.1145/3798129.3800922},
  eprint = {2512.14643},
  archivePrefix = {arXiv},
  note = {Full version arXiv:2512.14643v4, August 25, 2026}
}

@inproceedings{AnshuDongOuYao2025STOC,
  author        = {Anshu, Anurag and Dong, Yangjing and Ou, Fengning and Yao, Penghui},
  title         = {On the Computational Power of {$\mathsf{QAC}^{0}$} with Barely Superlinear Ancillae},
  booktitle     = {Proceedings of the 57th Annual ACM Symposium on Theory of Computing},
  series        = {STOC '25},
  year          = {2025},
  pages         = {1476--1487},
  publisher     = {Association for Computing Machinery},
  doi           = {10.1145/3717823.3718189},
  url           = {https://doi.org/10.1145/3717823.3718189}
}

@misc{NadimpalliParhamVasconcelosYuen2024,
  author        = {Nadimpalli, Shivam and Parham, Natalie and Vasconcelos, Francisca and Yuen, Henry},
  title         = {On the {Pauli} Spectrum of {$\mathsf{QAC}^{0}$}},
  year          = {2024},
  howpublished  = {arXiv:2311.09631},
  eprint        = {2311.09631},
  archivePrefix = {arXiv},
  primaryClass  = {quant-ph},
  doi           = {10.48550/arXiv.2311.09631},
  url           = {https://arxiv.org/abs/2311.09631v4},
  note          = {Version 4, submitted 17 July 2024; PDF dated 19 July 2024}
}

@article{FurstSaxeSipser1984,
  author = {Furst, Merrick and Saxe, James B. and Sipser, Michael},
  title = {Parity, circuits, and the polynomial-time hierarchy},
  journal = {Mathematical Systems Theory},
  volume = {17}, pages = {13--27}, year = {1984},
  doi = {10.1007/BF01744431},
  url = {https://doi.org/10.1007/BF01744431}
}
@article{Ajtai1983,
  author = {Ajtai, Mikl{\'o}s},
  title = {{$\Sigma^1_1$}-Formulae on Finite Structures},
  journal = {Annals of Pure and Applied Logic},
  volume = {24}, number = {1}, pages = {1--48}, year = {1983},
  doi = {10.1016/0168-0072(83)90038-6}
}
@inproceedings{Hastad1986,
  author = {H{\aa}stad, Johan},
  title = {Almost optimal lower bounds for small depth circuits},
  booktitle = {Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing},
  pages = {6--20}, year = {1986}, publisher = {Association for Computing Machinery},
  doi = {10.1145/12130.12132},
  url = {https://doi.org/10.1145/12130.12132}
}
@article{FangFennerGreenHomerZhang2006,
  author = {Fang, Maosen and Fenner, Stephen and Green, Frederic and Homer, Steven and Zhang, Yong},
  title = {Quantum lower bounds for fanout},
  journal = {Quantum Information and Computation},
  volume = {6}, number = {1}, pages = {46--57}, year = {2006},
  doi = {10.26421/QIC6.1-3},
  url = {https://www.rintonpress.com/journals/doi/QIC6.1-3.html}
}
@article{Bera2011,
  author = {Bera, Debajyoti},
  title = {A lower bound method for quantum circuits},
  journal = {Information Processing Letters},
  volume = {111}, number = {15}, pages = {723--726}, year = {2011},
  doi = {10.1016/j.ipl.2011.05.002},
  url = {https://doi.org/10.1016/j.ipl.2011.05.002}
}
@misc{PadeFennerGrierThierauf2020,
  author = {Pad{\'e}, Daniel and Fenner, Stephen and Grier, Daniel and Thierauf, Thomas},
  title = {Depth-2 {QAC} circuits cannot simulate quantum parity},
  year = {2020}, eprint = {2005.12169}, archivePrefix = {arXiv},
  primaryClass = {quant-ph}, url = {https://arxiv.org/abs/2005.12169v1},
  note = {Version 1, May 25, 2020}
}
@misc{FennerGrierPadeThierauf2025,
  author = {Fenner, Stephen and Grier, Daniel and Pad{\'e}, Daniel and Thierauf, Thomas},
  title = {Tight bounds on depth-2 {QAC}-circuits computing parity},
  year = {2025}, eprint = {2504.06433}, archivePrefix = {arXiv},
  primaryClass = {quant-ph}, url = {https://arxiv.org/abs/2504.06433v1},
  note = {Version 1, submitted April 8, 2025; PDF dated April 10, 2025}
}
@article{HoyerSpalek2005,
  author = {H{\o}yer, Peter and {\v S}palek, Robert},
  title = {Quantum Fan-out is Powerful},
  journal = {Theory of Computing},
  volume = {1}, number = {5}, pages = {81--103}, year = {2005},
  doi = {10.4086/toc.2005.v001a005},
  url = {https://theoryofcomputing.org/articles/v001a005/}
}

@article{AnshuMetger2023,
  author = {Anshu, Anurag and Metger, Tony},
  title = {Concentration bounds for quantum states and limitations on the {QAOA} from polynomial approximations},
  journal = {Quantum},
  volume = {7},
  pages = {999},
  year = {2023},
  doi = {10.22331/q-2023-05-11-999},
  url = {https://quantum-journal.org/papers/q-2023-05-11-999/}
}

@article{KuwaharaAradAmicoVedral2017,
  author = {Kuwahara, Tomotaka and Arad, Itai and Amico, Luigi and Vedral, Vlatko},
  title = {Local reversibility and entanglement structure of many-body ground states},
  journal = {Quantum Science and Technology},
  volume = {2},
  number = {1},
  pages = {015005},
  year = {2017},
  doi = {10.1088/2058-9565/aa523d},
  url = {https://arxiv.org/abs/1502.05330v3}
}

@article{Sherstov2009,
  author = {Sherstov, Alexander A.},
  title = {Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions},
  journal = {Computational Complexity},
  volume = {18},
  number = {2},
  pages = {219--247},
  year = {2009},
  url = {https://web.cs.ucla.edu/~sherstov/pdf/incl-excl.pdf}
}

@misc{OpenAILocalization2026,
  author = {{OpenAI}},
  title = {{Product-projection localization and the $\mathrm{QAC}^0$ parity lower bound}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026/paper.pdf}{OAI:Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026}},
  year = {2026}
}

@misc{Kintali2026ShallowParity,
  author = {Kintali, Shiva},
  title = {Parity in Shallow {QAC} Circuits: Correlation Decay and Exact Lower Bounds},
  year = {2026},
  month = sep,
  note = {Preprint dated September 5, 2026},
  url = {https://shivakintali.github.io/papers/QAC.pdf}
}

@misc{GGJ2026Fourier,
  author = {Gretta, Lucas and Gupta, Meghal and Joshi, Malvika Raj},
  title = {Parity $\notin$ {QAC0} $\iff$ {QAC0} is {Fourier}-Concentrated},
  year = {2026},
  eprint = {2604.02793},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  note = {Version 2, August 25, 2026},
  url = {https://arxiv.org/abs/2604.02793v2}
}

@misc{GrierMorrisWu2026,
  author = {Grier, Daniel and Morris, Jackson and Wu, Kewen},
  title = {{$\mathsf{QAC}^{0}$} Contains {$\mathsf{TC}^{0}$} (with Many Copies of the Input)},
  year = {2026},
  eprint = {2601.03243},
  archivePrefix = {arXiv},
  primaryClass = {cs.CC},
  note = {Version 1, January 6, 2026},
  url = {https://arxiv.org/abs/2601.03243v1}
}

@misc{XuLi2026Fanout,
  author = {Xu, Boyan and Li, Lvzhou},
  title = {Fanout Complexity of Symmetric {Boolean} Functions in {$\mathsf{QAC}^0$}},
  year = {2026},
  eprint = {2609.05153},
  archivePrefix = {arXiv},
  primaryClass = {quant-ph},
  note = {Version 1, September 4, 2026},
  url = {https://arxiv.org/abs/2609.05153v1}
}