Superpolynomial Circuits, Almost Sparse Oracles and the Exponential Hierarchy

Several problems concerning superpolynomial size circuits and superpolynomial-time advice classes are investigated. First we consider the implications of NP (and other fundamental complexity classes) having circuits slightly bigger than polynomial. We prove that if such circuits exist, for example if NP has n logn size circuits, the exponential hierarchy collapses to the second level. Next we consider the consequences of the bottom levels of the exponential hierarchy being contained in small advice classes. Again various collapses result. For example, if EXP NP is contained in EXP=poly then EXP NP = EXP. Finally, we consider the alternating 2 polylog-time hierarchy. The properties of this hierarchy underlie many of the previous results.