Document Open Access Logo

Quantum Merlin-Arthur and Proofs Without Relative Phase

Authors Roozbeh Bassirian, Bill Fefferman, Kunal Marwaha

Thumbnail PDF


  • Filesize: 1.3 MB
  • 19 pages

Document Identifiers

Author Details

Roozbeh Bassirian
  • University of Chicago, IL, USA
Bill Fefferman
  • University of Chicago, IL, USA
Kunal Marwaha
  • University of Chicago, IL, USA


Thanks to Zachary Remscrim for collaborating on early stages of this project. Thanks to Noam Lifshitz, Dor Minzer, and Kevin Pratt for answering questions about algebraic constructions of expanders. Thanks to Srinivasan Arunachalam, Fernando Granha Jeronimo, Supartha Podder, and Pei Wu for comments on a draft of this manuscript.

Cite AsGet BibTex

Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. Quantum Merlin-Arthur and Proofs Without Relative Phase. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [Grilo et al., 2014]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu [Jeronimo and Wu, 2023]) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some other constant gap is equal to QMA. One interpretation is that Merlin’s ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2) ⊆ NEXP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • quantum complexity
  • QMA(2)
  • PCPs


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Scott Aaronson. Open problems related to quantum query complexity, 2021. URL:
  2. Scott Aaronson, Salman Beigi, Andrew Drucker, Bill Fefferman, and Peter Shor. The power of unentanglement, 2008. URL:
  3. Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler. Quantum lower bounds for approximate counting via laurent polynomials, 2020. URL:
  4. Noga Alon. Explicit expanders of every degree and size. Combinatorica, pages 1-17, 2021. URL:
  5. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL:
  6. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs; A new characterization of NP. In 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, Pennsylvania, USA, 24-27 October 1992, pages 2-13. IEEE Computer Society, 1992. URL:
  7. Roozbeh Bassirian, Bill Fefferman, and Kunal Marwaha. On the power of nonstandard quantum oracles. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023), 2023. URL:
  8. Hugue Blier and Alain Tapp. A quantum characterization of np, 2010. URL:
  9. Anne Broadbent and Eric Culf. Rigidity for monogamy-of-entanglement games, 2023. URL:
  10. Samuel Burer. Copositive programming. In Handbook on semidefinite, conic and polynomial optimization, pages 201-218. Springer, 2011. URL:
  11. Jing Chen and Andrew Drucker. Short multi-prover quantum proofs for sat without entangled measurements, 2010. URL:
  12. Yuanyou Furui Cheng. Explicit estimate on primes between consecutive cubes, 2013. URL:
  13. Alessandro Chiesa and Michael A Forbes. Improved soundness for qma with multiple provers, 2011. URL:
  14. John F Clauser, Michael A Horne, Abner Shimony, and Richard A Holt. Proposed experiment to test local hidden-variable theories. Physical review letters, 23(15):880, 1969. URL:
  15. David Cui, Arthur Mehta, Hamoon Mousavi, and Seyed Sajjad Nezhadi. A generalization of CHSH and the algebraic structure of optimal strategies. Quantum, 4:346, October 2020. URL:
  16. Abhinav Deshpande, Alexey V Gorshkov, and Bill Fefferman. Importance of the spectral gap in estimating ground-state energies. PRX Quantum, 3(4):040327, 2022. URL:
  17. Irit Dinur. The pcp theorem by gap amplification. Journal of the ACM (JACM), 54(3):12-es, 2007. URL:
  18. François Le Gall, Shota Nakagawa, and Harumichi Nishimura. On QMA protocols with two short quantum proofs. Quantum Inf. Comput., 12(7-8):589-600, 2012. URL:
  19. Sevag Gharibian. Strong np-hardness of the quantum separability problem, 2009. URL:
  20. Sevag Gharibian, Miklos Santha, Jamie Sikora, Aarthi Sundaram, and Justin Yirka. Quantum generalizations of the polynomial hierarchy with applications to qma(2), 2018. URL:
  21. Alex B. Grilo, Iordanis Kerenidis, and Jamie Sikora. Qma with subset state witnesses, 2014. URL:
  22. Aram W. Harrow and Ashley Montanaro. Testing product states, quantum merlin-arthur games and tensor optimization. Journal of the ACM (JACM), 60(1):1-43, 2013. URL:
  23. Prahladh Harsha. Robust PCPs of proximity and shorter PCPs. PhD thesis, Massachusetts Institute of Technology, 2004. URL:
  24. Fernando Granha Jeronimo and Pei Wu. The power of unentangled quantum proofs with non-negative amplitudes. 55th Annual ACM Symposium on Theory of Computing, 2023. URL:
  25. Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. Mip*= re. Communications of the ACM, 64(11):131-138, 2021. URL:
  26. Hirotada Kobayashi, Keiji Matsumoto, and Tomoyuki Yamakami. Quantum merlin-arthur proof systems: Are multiple merlins more helpful to arthur?, 2003. URL:
  27. Greg Kuperberg. How hard is it to approximate the jones polynomial? Theory of Computing, 11(1):183-219, 2015. URL:
  28. Alexander Lubotzky. Finite simple groups of lie type as expanders, 2009. URL:
  29. Chris Marriott and John Watrous. Quantum arthur-merlin games, 2005. URL:
  30. Dominic Mayers and Andrew Yao. Self testing quantum apparatus, 2004. URL:
  31. Matthew McKague. On the power quantum computation over real hilbert spaces. International Journal of Quantum Information, 11(01):1350001, February 2013. URL:
  32. Anand Natarajan and Tina Zhang. Quantum free games. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 1603-1616, 2023. URL:
  33. Attila Pereszlényi. Multi-prover quantum merlin-arthur proof systems with small gap, 2012. URL:
  34. Ben W Reichardt, Falk Unger, and Umesh Vazirani. Classical command of quantum systems. Nature, 496(7446):456-460, 2013. URL:
  35. Adrian She and Henry Yuen. Unitary property testing lower bounds by polynomials, 2022. URL:
  36. Boris S Tsirelson. Some results and problems on quantum bell-type inequalities. Hadronic Journal Supplement, 8(4):329-345, 1993. URL:
  37. Jianwei Xu. Quantifying the phase of quantum states, 2023. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail