# Comparing SAT to MCSP reduction class separations and faster SAT class separations?

Assume $$SAT$$ is in $$QuasiP$$. We immediately infer $$NQuasiP=QuasiP$$ and $$EXP=NEXP$$. From https://people.csail.mit.edu/rrw/easy-witness-nqp.pdf we infer $$NQuasiP$$ is not in $$P/poly$$ which implies $$NQuasiP\neq ZPP$$.

Now if $$SAT$$ reduces $$MCSP$$ by $$LOGSPACE$$ $$m$$-reductions we infer a stronger separation of $$PSPACE\neq ZPP$$ by https://drops.dagstuhl.de/opus/volltexte/2015/5074/.

1. I am wondering if $$SAT$$ is in $$TISP(QuasiPTIME, QuasiLOGSPACE)$$ will provide $$PSPACE\not\subseteq P/poly$$ and $$PSPACE\neq ZPP$$?

I think at least $$SAT$$ in $$TISP(PTIME, LOGSPACE)$$ will provide $$PSPACE\not\subseteq P/poly$$ and $$PSPACE\neq ZPP$$.

1. Does faster $$SAT$$ provide implications to $$SAT$$ to $$MCSP$$ reductions or vice versa?

• SAT is in quasiP implies $NquasiP=QuasiP$ and $EXP=NEXP$ and doesn't it mean (utilizing time hierarchy) $QuasiP\subsetneq EXP=\Sigma_3EXP$? So is it possible following $$NP=CH\subseteq TISP(PTIME, LOGSPACE)\subsetneq QuasiP=TISP(QuasiPTIME, polylogspace))\subsetneq EXP=\Sigma_3EXP=PSPACE=TISP(EXPTIME, POLYSPACE))$$ is not unequivocally ruled out? Feb 12, 2021 at 15:58
• I guess $LOGSPACE$ containing $CH$ implies $EXP$ in $PSPACE$ is standard result. Feb 12, 2021 at 16:29