We present an alternate proof of the recent result by Gutfreund and Kawachi that derandomizing Arthur-Merlin games into P^{NP} implies linear-exponential circuit lower bounds for E^{NP}. Our proof is simpler and yields stronger results. In particular, consider the promise-AM problem of distinguishing between the case where a given Boolean circuit C accepts at least a given number b of inputs, and the case where C accepts less than \delta \cdot b inputs for some positive constant \delta. If P^{NP} contains a solution for this promise problem then E^{NP} requires circuits of size \Omega(2^n/n) almost everywhere.