{"author":[{"full_name":"Karaiskos, Georgios","last_name":"Karaiskos","first_name":"Georgios"},{"last_name":"Raza","first_name":"Asad","full_name":"Raza, Asad"},{"id":"57863","first_name":"Dorian","last_name":"Rudolph","full_name":"Rudolph, Dorian"},{"first_name":"Dax Enshan","last_name":"Koh","full_name":"Koh, Dax Enshan"},{"first_name":"Sevag","last_name":"Gharibian","orcid":"0000-0002-9992-3379","full_name":"Gharibian, Sevag","id":"71541"}],"status":"public","title":"Verification Complexity and Extension of Classical Shadows","year":"2026","date_updated":"2026-10-01T07:33:38Z","language":[{"iso":"eng"}],"_id":"67292","user_id":"71541","citation":{"ieee":"G. Karaiskos, A. Raza, D. Rudolph, D. E. Koh, and S. Gharibian, “Verification Complexity and Extension of Classical Shadows,” <i>arXiv:2609.40107</i>. 2026.","apa":"Karaiskos, G., Raza, A., Rudolph, D., Koh, D. E., &#38; Gharibian, S. (2026). Verification Complexity and Extension of Classical Shadows. In <i>arXiv:2609.40107</i>.","short":"G. Karaiskos, A. Raza, D. Rudolph, D.E. Koh, S. Gharibian, ArXiv:2609.40107 (2026).","chicago":"Karaiskos, Georgios, Asad Raza, Dorian Rudolph, Dax Enshan Koh, and Sevag Gharibian. “Verification Complexity and Extension of Classical Shadows.” <i>ArXiv:2609.40107</i>, 2026.","mla":"Karaiskos, Georgios, et al. “Verification Complexity and Extension of Classical Shadows.” <i>ArXiv:2609.40107</i>, 2026.","bibtex":"@article{Karaiskos_Raza_Rudolph_Koh_Gharibian_2026, title={Verification Complexity and Extension of Classical Shadows}, journal={arXiv:2609.40107}, author={Karaiskos, Georgios and Raza, Asad and Rudolph, Dorian and Koh, Dax Enshan and Gharibian, Sevag}, year={2026} }","ama":"Karaiskos G, Raza A, Rudolph D, Koh DE, Gharibian S. Verification Complexity and Extension of Classical Shadows. <i>arXiv:260940107</i>. Published online 2026."},"publication":"arXiv:2609.40107","abstract":[{"text":"Classical shadows are an influential framework for compressing copies of a given quantum state $ρ$ into classical data $S$, enabling many properties of $ρ$ to be predicted from relatively few copies. In this work, we study two natural questions involving shadows: (1) Given $S$, when can one efficiently verify that $S$ came from a genuine $n$-qubit state? This is called the Classical Shadow Validity (CSV) problem, introduced by Karaiskos, Rudolph, Meyer, Eisert, and Gharibian [ICALP 2026]. (2) Given $S$ that allows one to capture 2-local properties of $ρ$, can one fake or spoof a shadow $S'$ which predicts 3-local properties of some state? For (1), we show CSV is efficiently solvable for permutation-invariant shadows, QMA-hard for real, fermionic, and bosonic shadows, and both coNP-hard and QMA-hard when the observable family consists of all $n$-qubit Pauli strings. A result of independent interest along the way is a new upper bound qc-$Σ_2$ $\\subseteq$ $\\mathrm{P}^{\\mathrm{PP}}$, where qc-$Σ_2$ is a quantum analogue of the second level of the polynomial hierarchy in which the first proof is quantum. For (2), we show intractability: Given the 2-local marginals $S$ of a quantum state $ρ$, estimating the 3-local marginals of $ρ$ is intractable unless QCMA $\\subseteq$ BPP, even if the state $ρ$ is the unique state consistent with $S$.","lang":"eng"}],"date_created":"2026-10-01T07:32:45Z","external_id":{"arxiv":["2609.40107"]},"department":[{"_id":"7"},{"_id":"623"}],"type":"preprint"}