Quasipolynomial and polynomial time hierarchy
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.