It is specific to cases where a quantum algorithm exists that provides speedup, it is not at all generic. The complexity class of interest is BQP: https://en.wikipedia.org/wiki/BQP

Also of note: P is in BQP, but it is not proven that BQP != P. Some problems like factoring have a known polynomial time algorithm, and the best known classical algorithm is exponential, which is where you see these massive speedups. But we don't know that there isn't an unknown polynomial time classical factoring algorithm and we just haven't discovered it yet. It is a (widely believed) conjecture, that there are hard problems solved in BQP that are outside P.