Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)

S. Gharibian, M. Santha, J. Sikora, A. Sundaram, J. Yirka, in: I. Potapov, P. Spirakis, J. Worrell (Eds.), 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018), Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018, pp. 58:1-58:16.

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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 does not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH, uses classical proofs, and the second, 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, Q Sigma_3, into NEXP using the Ellipsoid Method for efficiently solving semidefinite programs. These results yield two implications for QMA(2), the variant of Quantum Merlin-Arthur (QMA) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if 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 P^{PP^{PP}}). Second, unless QMA(2)= Q Sigma_3 (i.e., alternating quantifiers do not help in the presence of "unentanglement"), QMA(2) is strictly contained in NEXP.
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Proceedings Title
43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
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117
Page
58:1-58:16
Conference
43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
Conference Location
Liverpool, UK
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Gharibian S, Santha M, Sikora J, Sundaram A, Yirka J. Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2). In: Potapov I, Spirakis P, Worrell J, eds. 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018). Vol 117. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik; 2018:58:1-58:16. doi:10.4230/LIPIcs.MFCS.2018.58
Gharibian, S., Santha, M., Sikora, J., Sundaram, A., & Yirka, J. (2018). Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2). In I. Potapov, P. Spirakis, & J. Worrell (Eds.), 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018) (Vol. 117, pp. 58:1-58:16). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.MFCS.2018.58
@inproceedings{Gharibian_Santha_Sikora_Sundaram_Yirka_2018, place={Dagstuhl, Germany}, series={Leibniz International Proceedings in Informatics (LIPIcs)}, title={Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)}, volume={117}, DOI={10.4230/LIPIcs.MFCS.2018.58}, booktitle={43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)}, publisher={Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik}, author={Gharibian, Sevag and Santha, Miklos and Sikora, Jamie and Sundaram, Aarthi and Yirka, Justin}, editor={Potapov, Igor and Spirakis, Paul and Worrell, JamesEditors}, year={2018}, pages={58:1-58:16}, collection={Leibniz International Proceedings in Informatics (LIPIcs)} }
Gharibian, Sevag, Miklos Santha, Jamie Sikora, Aarthi Sundaram, and Justin Yirka. “Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2).” In 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018), edited by Igor Potapov, Paul Spirakis, and James Worrell, 117:58:1-58:16. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. https://doi.org/10.4230/LIPIcs.MFCS.2018.58.
S. Gharibian, M. Santha, J. Sikora, A. Sundaram, and J. Yirka, “Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2),” in 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018), Liverpool, UK, 2018, vol. 117, pp. 58:1-58:16.
Gharibian, Sevag, et al. “Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2).” 43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018), edited by Igor Potapov et al., vol. 117, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018, pp. 58:1-58:16, doi:10.4230/LIPIcs.MFCS.2018.58.

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