Quantum generalizations of the polynomial hierarchy with applications to QMA(2)
Bibliographic record
Abstract
The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH do not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, $$\rm{QCPH}$$ , uses classical proofs, and the second, $$\rm{QPH}$$ , uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, $$\rm{Q\Sigma_3}$$ , into NEXP using the ellipsoid method for efficiently solving semidefinite programs. These results yield two implications for $$\rm{QMA(2)}$$ , the variant of Quantum Merlin-Arthur ( $$\rm{QMA}$$ ) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if $$\rm{QCPH = QPH}$$ (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs ``equivalent''), then QMA(2) is in the counting hierarchy (specifically, in $${\rm P}^{{\rm pp}^{{\rm pp}}}$$ ). Second, because $$\rm{QMA(2)}\subseteq \rm{Q\Sigma_3}$$ , $$\rm{QMA(2)}$$ is strictly contained in NEXP unless $$\rm{QMA(2)}=\rm{Q\Sigma_3}$$ (i.e., alternating quantifiers do not help in the presence of ``unentanglement'').
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".