Time-Space Tradeoffs in the Counting Hierarchy

We extend the lower bound techniques of [17], to the unbounded-error probabilistic model. A key step in the argument is a generalization of Nepomnjaščiı̆’s theorem from the Boolean setting to the arithmetic setting. This generalization is made possible, due to the recent discovery of logspaceuniform TC0 circuits for iterated multiplication [11]. Here is an example of the sort of lower bounds that we obtain: we show that MAJ·MAJSAT is not contained in PrTiSp(n1+o(1), n ) for any < 1. We also extend a lower bound of [17], from showing that SAT does not have uniform NC1 circuits of size n1+o(1), to a similar result for SAC1 circuits.