Quadratic Simulations of Merlin–Arthur Games

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).