PapersIt is well-known that $\mathsf{AC^0}$ circuits cannot compute inner product (since parity is hard for $\mathsf{AC^0}$).
What if we allow each of the parties to preprocess their input?
If we could show that bounded depth circuits of quasipolynomial size cannot computer inner product even with arbitrary polynomial length preprocessing, then this would imply that inner product is outside the polynomial hierarchy of communication complexity ($\mathsf{PH^{cc}}$).
We show that $\mathsf{AC^0}$ circuits cannot compute inner product even if one party has unlimited preprocessing, and the other one is limited to preprocessing of length $n+n/\log^{\omega(1)} n$.
Our lower bound also applies to pseudorandom functions.
In ongoing work, we extend these results to correlation bounds.
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