  We prove a downward separation for $\mathsf{\Sigma}_2$-time classes.
Specifically, we prove that if $\Sigma_2$E does not have polynomial size
non-deterministic circuits, then $\Sigma_2$SubEXP does not have \textit{fixed}
polynomial size non-deterministic circuits. To achieve this result, we use
Santhanam's technique on augmented Arthur-Merlin protocols defined by
Aydinlio\u{g}lu and van Melkebeek. We show that augmented Arthur-Merlin
protocols with one bit of advice do not have fixed polynomial size
non-deterministic circuits. We also prove a weak unconditional derandomization
of a certain type of promise Arthur-Merlin protocols. Using Williams' easy
hitting set technique, we show that $\Sigma_2$-promise AM problems can be
decided in $\Sigma_2$SubEXP with $n^c$ advice, for some fixed constant $c$.
