<p>This is not currently known. In <a href="https://theses.liacs.nl/pdf/2017-2018-VinkhuijzenLieuwe.pdf" rel="nofollow noreferrer">my MSc. thesis</a>, I show that, if it were true, then a consequence would be that <span class="math-container">$NP^{NP}\subseteq QMA$</span> (Theorem 21, below). I conjecture that in fact <span class="math-container">$coNP\subseteq 
 QMA$</span> implies <span class="math-container">$PH\subseteq QMA$</span>, but I was not able to prove that.</p>
<p><strong>Lemma 14.</strong> <span class="math-container">$NP^{QMA\cap coQMA}\subseteq QMA$</span>.</p>
<p><strong>Theorem 21.</strong> If <span class="math-container">$QMA$</span> contains <span class="math-container">$co\text-NP$</span>, then <span class="math-container">$NP^{NP}\subseteq QMA$</span></p>
<p><em>Proof.</em> If <span class="math-container">$QMA$</span> contains <span class="math-container">$coNP$</span>, then, equivalently, <span class="math-container">$NP\subseteq coQMA$</span>, so we have <span class="math-container">$NP\subseteq QMA\cap coQMA$</span>. Plugging this in, we get <span class="math-container">$NP^{NP}\subseteq NP^{QMA\cap coQMA}\subseteq QMA$</span> by Lemma 14. <span class="math-container">$\square$</span></p>
<p>The thesis contains other consequences, and variations and generalizations of this theorem.</p>

CC BY-SA 4.0
2021-08-27T19:30:48.403Z
2021-08-27T19:30:48.403Z
Lieuwe Vinkhuijzen
<p>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.)</p>

CC BY-SA 4.0
2021-08-16T05:13:22.110Z
Not yet: https://cs.stackexchange.com/questions/128509/is-qma-known-to-contain-co-np
Avi Tal
CC BY-SA 4.0
2021-08-18T08:57:49.067Z
I have answered the original question on cs stackexchange. Should this question be closed now?
Lieuwe Vinkhuijzen
<cc.complexity-theory><np-hardness><complexity-classes><complexity><qma>
Is QMA known to contain Co-NP?
blademan9999
# Is QMA known to contain Co-NP?

(Asked by: username_0 on 2021-08-16)

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.)

**Tags:** cc.complexity-theory, np-hardness, complexity-classes, complexity, qma
> **Comments:**
> **username_1**: Not yet: https://cs.stackexchange.com/questions/128509/is-qma-known-to-contain-co-np  
> **username_2**: I have answered the original question on cs stackexchange. Should this question be closed now?  

---

## 1 answers

### Answer by username_2 on 2021-08-27. Score: 4 | Accepted answer

This is not currently known. In [my MSc. thesis](https://theses.liacs.nl/pdf/2017-2018-VinkhuijzenLieuwe.pdf), 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. $\square$

The thesis contains other consequences, and variations and generalizations of this theorem.


---

