A recent and active line of work achieves tight lower bounds for fundamental problems under the Strong Exponential Time Hypothesis (SETH). A celebrated result of Backurs and Indyk (STOC'15) proves that the Edit Distance of two sequences of length n cannot be computed in strongly subquadratic time under SETH. The result was extended by follow-up works to simpler looking problems like finding the Longest Common Subsequence (LCS). SETH is a very strong assumption, asserting that even linear size CNF formulas cannot be analyzed for satisfiability with an exponential speedup over exhaustive search. We consider much safer assumptions, e.g. that such a speedup is impossible for SAT on much more expressive representations, like NC circuits. Intuitively, this seems much more plausible: NC circuits can implement complex cryptographic primitives, while CNFs cannot even approximately compute an XOR of bits. Our main result is a surprising reduction from SAT on Branching Programs to fundamental problems in P like Edit Distance, LCS, and many others. Truly subquadratic algorithms for these problems therefore have consequences that we consider to be far more remarkable than merely faster CNF SAT algorithms. For example, SAT on arbitrary o(n)-depth bounded fan-in circuits (and therefore also NC-Circuit-SAT) can be solved in (2-eps)^n time. A very interesting feature of our work is that we can prove major consequences even from mildly subquadratic algorithms for Edit Distance or LCS. For example, we show that if we can shave an arbitrarily large polylog factor from n^2 for Edit Distance then NEXP does not have non-uniform NC^1 circuits. A more fine-grained examination shows that even shaving a $\log^c{n}$ factor, for a specific constant $c \approx 10^3$, already implies new circuit lower bounds.