<p>I am not conversant in the complexity class <span class="math-container">$P/poly$</span>. While reading about the class on wiki I encountered two conditional statements about it, namely:</p>
<ol>
<li><p>If <span class="math-container">$NP ⊆ P/poly$</span> then <span class="math-container">$PH$</span> (the polynomial hierarchy) collapses to <span class="math-container">${\displaystyle \Sigma _{2}^{\mathsf {P}}}\cap{\displaystyle \Sigma _{2}^{\mathsf {P}}}$</span>. This result is the Karp–Lipton theorem; furthermore, <span class="math-container">$NP ⊆ P/poly$</span> implies <span class="math-container">$AM = MA$</span></p>
</li>
<li><p>If <span class="math-container">$PSPACE ⊆ P/poly$</span> then <span class="math-container">${\displaystyle {\mathsf {PSPACE}}=\Sigma _{2}^{\mathsf {P}}\cap \Pi _{2}^{\mathsf {P}}}$</span> , even <span class="math-container">$PSPACE = MA$</span>.</p>
</li>
</ol>
<p><strong>Query 1:</strong> Let us assume <span class="math-container">${\displaystyle {\mathsf {PSPACE}}=\Sigma _{2}^{\mathsf {P}}}$</span> and the hierarchy collapses. Does the opposite also holds true, i.e. does that imply: <span class="math-container">$PSPACE ⊆ P/poly$</span> and <span class="math-container">$PSPACE = MA$</span>?</p>
<p><strong>Query 2:</strong> Assuming the answer to 1. is No, does this collapse clarify anything (more than the already known relationships) b/w <span class="math-container">${\mathsf {PSPACE}}/\Sigma _{2}^{\mathsf {P}}$</span> and other 'lower' complexity classes <span class="math-container">$MA$</span>, <span class="math-container">$AM$</span>, <span class="math-container">$QMA$</span> <span class="math-container">$BQP$</span> or <span class="math-container">$P/poly$</span>?</p>

CC BY-SA 4.0
2021-10-31T13:07:29.040Z
In general, knowing that two classes high in the hierarchy are equal, does not have implications about the lower parts. Of course, it is possible that there are exceptions to this rule.
domotorp
CC BY-SA 4.0
2021-11-01T05:15:58.240Z
thanks. I understand. And in the case mentioned above (as of the knowledge we have now) we won't know anything more about the relationship b/w two sub branches below, even in the event of this collapse: namely NP, MA, AM and PSPACE and the other being the quantum ones BQP and QMA? (I am guessing that is the case, but just reconfirming)
TheoryQuest1
<cc.complexity-theory><complexity-classes>
Query about P/poly and Polynomial Hierarchy Collapse to $\Sigma _{2}$
TheoryQuest1
# Query about P/poly and Polynomial Hierarchy Collapse to $\Sigma _{2}$

(Asked by: username_0 on 2021-10-31)

I am not conversant in the complexity class $P/poly$. While reading about the class on wiki I encountered two conditional statements about it, namely:

1. If $NP ⊆ P/poly$ then $PH$ (the polynomial hierarchy) collapses to ${\displaystyle \Sigma \_{2}^{\mathsf {P}}}\cap{\displaystyle \Sigma \_{2}^{\mathsf {P}}}$. This result is the Karp–Lipton theorem; furthermore, $NP ⊆ P/poly$ implies $AM = MA$
2. If $PSPACE ⊆ P/poly$ then ${\displaystyle {\mathsf {PSPACE}}=\Sigma \_{2}^{\mathsf {P}}\cap \Pi \_{2}^{\mathsf {P}}}$ , even $PSPACE = MA$.

**Query 1:** Let us assume ${\displaystyle {\mathsf {PSPACE}}=\Sigma \_{2}^{\mathsf {P}}}$ and the hierarchy collapses. Does the opposite also holds true, i.e. does that imply: $PSPACE ⊆ P/poly$ and $PSPACE = MA$?

**Query 2:** Assuming the answer to 1. is No, does this collapse clarify anything (more than the already known relationships) b/w ${\mathsf {PSPACE}}/\Sigma \_{2}^{\mathsf {P}}$ and other 'lower' complexity classes $MA$, $AM$, $QMA$ $BQP$ or $P/poly$?

**Tags:** cc.complexity-theory, complexity-classes

> **Comments:**
> **username_1**: In general, knowing that two classes high in the hierarchy are equal, does not have implications about the lower parts. Of course, it is possible that there are exceptions to this rule.  
> **username_0**: thanks. I understand. And in the case mentioned above (as of the knowledge we have now) we won't know anything more about the relationship b/w two sub branches below, even in the event of this collapse: namely NP, MA, AM and PSPACE and the other being the quantum ones BQP and QMA? (I am guessing that is the case, but just reconfirming)  

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## 0 answers

