Quadratic Simulations of Merlin–Arthur Games

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


The known proofs of \(\textsf {MA}\subseteq \textsf {PP}\) incur a quadratic overhead in the running time. We prove that this quadratic overhead is necessary for black-box simulations; in particular, we obtain an oracle relative to which \({\textsf {MA}\text {-}\textsf {TIME}(t)\not \subseteq \textsf {P}\text {-}\textsf {TIME}(o(t^2))}\). We also show that 2-sided-error Merlin–Arthur games can be simulated by 1-sided-error Arthur–Merlin games with quadratic overhead. We also present a simple, query complexity based proof (provided by Mika Göös) that there is an oracle relative to which \(\textsf {MA}\not \subseteq \textsf {NP}^\textsf {BPP}\) (which was previously known to hold by a proof using generics).



I thank Mika Göös for suggesting the proof of Lemma 5, and anonymous reviewers for helpful comments.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of MemphisMemphisUSA

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