---
res:
  bibo_abstract:
  - '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 $S$.@eng'
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Georgios
      foaf_name: Karaiskos, Georgios
      foaf_surname: Karaiskos
  - foaf_Person:
      foaf_givenName: Asad
      foaf_name: Raza, Asad
      foaf_surname: Raza
  - foaf_Person:
      foaf_givenName: Dorian
      foaf_name: Rudolph, Dorian
      foaf_surname: Rudolph
      foaf_workInfoHomepage: http://www.librecat.org/personId=57863
  - foaf_Person:
      foaf_givenName: Dax Enshan
      foaf_name: Koh, Dax Enshan
      foaf_surname: Koh
  - foaf_Person:
      foaf_givenName: Sevag
      foaf_name: Gharibian, Sevag
      foaf_surname: Gharibian
      foaf_workInfoHomepage: http://www.librecat.org/personId=71541
    orcid: 0000-0002-9992-3379
  dct_date: 2026^xs_gYear
  dct_language: eng
  dct_title: Verification Complexity and Extension of Classical Shadows@
...