Derandomization techniques are used to show that at least
one of the following holds regarding the size of the counting
complexity class SPP.
SPP has p-measure 0.
PH is contained in SPP.
In other words, SPP is small by being a negligible subset of
exponential time or large by containing the entire polynomial-time
hierarchy. This addresses an open problem about the complexity of the
graph isomorphism problem: it is not weakly complete for exponential
time unless PH is contained in SPP. It is also shown that the
polynomial-time hierarchy is contained in SPPNP if NP does
not have p-measure 0.