We may be able to solve SAT in $c^{\sqrt{n}}$ for $n$ variables and a constant $c$. Now we can suppose that we require a certain amount of preprocessing to get this result. For example, in this answer, Ryan Williams points out that we could have a large amount of preprocessing, and SAT in QP (quasi-polynomial) would still be interesting.
So my question is, if we solve SAT in $c^{\sqrt{n}}$, how much preprocessing can we have and still have an interesting result? In other words, if we had something like $2^{2^n}$ preprocessing, would this remain an interesting result? Basically, we would be saying that SAT has $c^{\sqrt{n}}$ sized circuits.
