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

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

2. a\cdot NEXP^{RP} does not have poly size constant-free arithmetic circuits.

3. For every fixed k, a\cdot MA does not have arithmetic circuits of size n^k.

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[2^{n^{o(1)}}]/n^{o(1)} if and only if there exists a family of multilinear polynomials in a\cdot NE/lin that requires constant-free arithmetic circuits of super-polynomial size and formal degree.