We associate to each Boolean language complexity class the algebraic class a consisting of families of polynomials fn for which the evaluation problem over the integers is in . We prove the following lower bound and randomness-to-hardness results:

1. If polynomial identity testing (PIT) is in NSUBEXP then aNEXP does not have poly size constant-free arithmetic circuits.

2. aNEXPRP does not have poly size constant-free arithmetic circuits.

3. For every fixed k, aMA does not have arithmetic circuits of size nk.

Items 1 and 2 strengthen two results due to Kabanets and Impagliazzo. The third item improves a lower bound due to Santhanam.

We consider the special case low-PIT of identity testing for (constant-free) arithmetic circuits with low formal degree, and give improved hardness-to-randomness trade-offs that apply to this case.

Combining our results for both directions of the hardness-randomness connection, we demonstrate a case where derandomization of PIT and proving lower bounds are equivalent. Namely, we show that low-PIT is infinitely often in NTIME[2no(1)]no(1) if and only if there exists a family of multilinear polynomials in aNElin that requires constant-free arithmetic circuits of super-polynomial size and formal degree