Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers

Authors Abhijit S. Mudigonda , R. Ryan Williams

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Abhijit S. Mudigonda
  • Portland, OR, USA
R. Ryan Williams
  • EECS and CSAIL, MIT, Cambridge, MA, USA


We thank the anonymous reviewers for helpful comments. The first author thanks Ryan Williams for his support and patience throughout this research. The first author also thanks Aram Harrow, Peter Shor, Saeed Mehraban, Ashwin Sah, and Lisa Yang for contributing office space and helpful conversations and Lijie Chen and Shyan Akmal for reading and editing a draft of this manuscript. Lastly, the first author apologizes to the Theory Group at MIT CSAIL eating so many of the chocolate-covered pretzels.

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Abhijit S. Mudigonda and R. Ryan Williams. Time-Space Lower Bounds for Simulating Proof Systems with Quantum and Randomized Verifiers. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the SAT problem and related problems within the polynomial-time hierarchy. For example, for the SAT problem, the state-of-the-art is that the problem cannot be solved by random-access machines in n^c time and n^o(1) space simultaneously for c < 2cos(π/7) ≈ 1.801. We extend this lower bound approach to the quantum and randomized domains. Combining Grover’s algorithm with components from SAT time-space lower bounds, we show that there are problems verifiable in O(n) time with quantum Merlin-Arthur protocols that cannot be solved in n^c time and n^o(1) space simultaneously for c < (3+√3)/2 ≈ 2.366, a super-quadratic time lower bound. This result and the prior work on SAT can both be viewed as consequences of a more general formula for time lower bounds against small-space algorithms, whose asymptotics we study in full. We also show lower bounds against randomized algorithms: there are problems verifiable in O(n) time with (classical) Merlin-Arthur protocols that cannot be solved in n^c randomized time and O(log n) space simultaneously for c < 1.465, improving a result of Diehl. For quantum Merlin-Arthur protocols, the lower bound in this setting can be improved to c < 1.5.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Time-space tradeoffs
  • lower bounds
  • QMA


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