Is there anyone prove the results such like the follows?
If $NP\not\subseteq BP(2^{\Omega(n)}),$ then $BPP\subseteq P^{NP}$
In summary, my question it that, can we get some derandomized results based on some nondeterminitic assumptions.
Is there anyone prove the results such like the follows? If $NP\not\subseteq BP(2^{\Omega(n)}),$ then $BPP\subseteq P^{NP}$ In summary, my question it that, can we get some derandomized results based on some nondeterminitic assumptions. 


As for derandomization under nondeterministic assumptions, you can basically relativize the usual results such as ImpagliazzoWigderson. Directly, this gives: if some language in $E^{NP}$ requires exponential circuits with an NPoracle, then $BPP^{NP}=P^{NP}$. There are similar results by Miltersen and Vinodchandran: if some language in $NE\cap coNE$ requires exponentialsize nondeterministic circuits, then $AM = NP$. 

