# Is QMA known to contain Co-NP?

Is QMA known to contain Co-NP? If not, would Co-NP being contained in QMA have any implications for other complexity classes. (e.g. Causing the polynomial heirachy to collapse.)

• Aug 16 at 5:13
• I have answered the original question on cs stackexchange. Should this question be closed now? Aug 18 at 8:57

This is not currently known. In my MSc. thesis, I show that, if it were true, then a consequence would be that $$NP^{NP}\subseteq QMA$$ (Theorem 21, below). I conjecture that in fact $$coNP\subseteq QMA$$ implies $$PH\subseteq QMA$$, but I was not able to prove that.
Lemma 14. $$NP^{QMA\cap coQMA}\subseteq QMA$$.
Theorem 21. If $$QMA$$ contains $$co\text-NP$$, then $$NP^{NP}\subseteq QMA$$
Proof. If $$QMA$$ contains $$coNP$$, then, equivalently, $$NP\subseteq coQMA$$, so we have $$NP\subseteq QMA\cap coQMA$$. Plugging this in, we get $$NP^{NP}\subseteq NP^{QMA\cap coQMA}\subseteq QMA$$ by Lemma 14. $$\square$$