Quasipolynomial and polynomial time hierarchy Consider the following two scenarios:


$NP\subseteq RTIME[2^{(\log n)^c}]$ or $NP\subseteq DTIME[2^{(\log n)^c}]$, i.e., $NP$ can be solved in quasipolynomial randomized or deterministic time. 
$NP$ has quasipolynomial size circuits.


Does the polynomial time hierarchy $PH$ collapse under any of the above conditions?

The first condition implies $EXP=NEXP$. Does the second imply the same or something weaker?

Could $P=BPP$ still hold in such situations?



Separately what implications follow from $EXP=NEXP$ and/or $EXP$ having quasipolynomial circuits?
If $NP$ can be solved in quasipolynomial time, then $NP$ has quasi polynomial size circuit. 
The second condition implies that the exponential time hierarchy collapses to the second level (as was shown in the paper of Buhrman and Homer). It was improved the collapse to $S_2^{exp}$ in a paper of Pavan et al.
As far as I know, these hypotheses do not easily imply a collapse of PH.