Couldn't find this one anywhere... It's an open problem whether $\Sigma_2 EXP$ problems have exponential-size circuit complexity. Is there an oracle relative to which $\Sigma_2 EXP$ has $2^{o(n)}$ size circuits? $2^{n^{o(1)}}$ size?Such an oracle exists for the class$MAEXP$; this was shown by Buhrman, Fortnow, and Thierauf. 
Oracles for sub-exponential circuit complexity of $\Sigma_2 EXP$
circuit complexity;oracles;alternation