What is the minimal time until a quantum system undergoing unitary dynamics can exhibit genuine quantum features? To answer this question we derive quantum speed limits (QSLs) for two-time correlation functions arising from statistics of measurements. These two-time correlators are described by Kirkwood-Dirac quasiprobabilities, if the initial quantum state of the system does not commute with the measurement observables. The QSLs here introduced are derived from the Schrödinger-Robertson uncertainty relation, and set the minimal time at which the real part of a quasiprobability can become negative and the corresponding imaginary part can be different from zero or crosses a given threshold. This departure of Kirkwood-Dirac quasiprobabilities from positivity is evidence for the onset of non-classical traits in the quantum dynamics. As an illustrative example, we apply these results to a conditional quantum gate by determining the optimal condition that gives rise to non-classicality at maximum speed. In this way, our analysis hints at boosted power extraction due to genuinely non-classical dynamics.