@inproceedings{8161,
  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.}},
  author       = {{Gharibian, Sevag and Santha, Miklos and Sikora, Jamie and Sundaram, Aarthi and Yirka, Justin}},
  booktitle    = {{43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)}},
  editor       = {{Potapov, Igor and Spirakis, Paul and Worrell, James}},
  keywords     = {{Complexity Theory, Quantum Computing, Polynomial Hierarchy, Semidefinite Programming, QMA(2), Quantum Complexity}},
  location     = {{Liverpool, UK}},
  pages        = {{58:1--58:16}},
  publisher    = {{Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik}},
  title        = {{{Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)}}},
  doi          = {{10.4230/LIPIcs.MFCS.2018.58}},
  volume       = {{117}},
  year         = {{2018}},
}

@article{8171,
  abstract     = {{The polynomial hierarchy plays a central role in classical complexity theory. Here, we define
a quantum generalization of the polynomial hierarchy, and initiate its study. We show that
not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact hard to approximate. Using the same techniques, we
also obtain hardness of approximation for the class QCMA. Our approach is based on the
use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy [Umans, FOCS 1999].
The problems for which we prove hardness of approximation for include, among others, a
quantum version of the Succinct Set Cover problem, and a variant of the local Hamiltonian
problem with hybrid classical-quantum ground states.}},
  author       = {{Gharibian, Sevag and Kempe, Julia}},
  journal      = {{Quantum Information & Computation}},
  keywords     = {{Hardness of approximation, polynomial time hierarchy, succinct set cover, quantum complexity}},
  number       = {{5-6}},
  pages        = {{517--540}},
  title        = {{{Hardness of approximation for quantum problems}}},
  volume       = {{14}},
  year         = {{2014}},
}