# Evidence for $\mathsf{P} \neq \mathsf{PP}$ if the polynomial hierarchy collapses?

We think that $$\mathsf{PH}$$ does not collapse, and that $$\mathsf{PP}$$ is not in $$\mathsf{P}$$.

Suppose on the contrary that $$\mathsf{PH}$$ does collapse, say even $$\mathsf{P}= \mathsf{NP}$$.

$$\mathsf{PP}$$ still seems like it would be of reach.

What evidence do we have that $$\mathsf{P} \neq \mathsf{PP}$$, under that assumption, i.e., evidence that does not apply to $$\mathsf{P}$$ vs. $$\mathsf{NP}$$?

$$\underline{\mbox{Assumption}}$$: Note if $$\mathsf{SAT}$$ is in linear time then perhaps $$\mathsf{P=PSPACE}$$ might follow as composition of linear time is linear time?

$$\underline{\mbox{Following is only based on if above assumption follows}}$$:

1. Perhaps the problem would be

1a. why believe $$\mathsf{SAT}$$ is in superlinear time

1b. and if there is reason to believe $$\mathsf{SAT}$$ is in superlinear time then why believe $$\mathsf{PP}$$ requires $$\Omega(\log\log n)$$ quantifications?

2. Do evidences provided by https://eccc.weizmann.ac.il/report/2018/107/ and http://arxiv.org/abs/1504.03398 still hold up in face of $$\mathsf{P=NP}$$ and $$\mathsf{SAT}$$ in linear time ($$\mathsf{P=NP}$$ seems to invalidate second)?

• Why the downvotes? Am I missing something here? Sep 9, 2019 at 3:46
• Is it plausible that $\mathsf{PH}$ collapses but that $\mathsf{P}\neq \mathsf{\# P}$? That sounds more tractable and an affirmative answer would be an affirmative answer to this question. Sep 11, 2019 at 16:49
• @JoshuaGrochow Would your comment in cstheory.stackexchange.com/q/198 and relations among $P$, $NP$, $NL$, $GapL$ and $GapP$ address $P=NP\neq PP$? Sep 15, 2019 at 8:56

The scaled down version of $$\mathsf{PH}$$ versus $$\mathsf{PP}$$ is $$\mathsf{AC}^0$$ versus $$MAJ \circ \mathsf{AC}^0$$, and we know that for the latter there is an exponential separation. Of course, this separation doesn't propagate exponentially up, but you could take this as philosophical evidence that $$\mathsf{PH}$$ is different enough from $$\mathsf{PP}$$ that perhaps that remains true even if $$\mathsf{P} = \mathsf{NP} = \mathsf{PH}$$. (In fact, following this line of reasoning, I'd guess you can build an oracle relative to which $$\mathsf{P} = \mathsf{NP} \neq \mathsf{PP}$$.)
• I think $\mathsf{PP}$ corresponds to $MAJ \circ polylog$ where $polylog$ is any gate with polylogarithmic fan-in. And we know that $MAJ$ itself is already not in $\mathsf{AC}^0$. Therefore scaling up we get $\mathsf{PP} \not \subseteq \mathsf{PH}$ in some relativized world. Oct 28, 2019 at 13:26