What do stronger circuit lower bounds give in terms of derandomization?
It is known that if $E = DTIME(2^{O(n)})$ is not contained in $SIZE(2^{\varepsilon \cdot n})$ for some $\varepsilon&gt;0$  then $BPP = P$ (https://dl.acm.org/citation.cfm?id=258590).

(Actually, a slightly stronger assumption is needed, namely, the separation between $E$ and $SIZE(2^{\varepsilon \cdot n})$ should hold for almost all input lengths - thanks to Ryan O'Donnel for pointing it out.)

More generally, this paper gives a direct optimal translation of any circuit lower bound into a corresponding derandomization result:
http://users.cms.caltech.edu/~umans/papers/SU01-final.pdf