  What is the minimal time until a quantum system can exhibit genuine quantum
features? To answer this question we derive quantum speed limits for two-time
correlation functions arising from statistics of measurements. Generally, these
two-time correlators are described by quasiprobabilities, if the initial
quantum state of the system does not commute with the measurement observables.
Our quantum speed limits are derived from the Heisenberg-Robertson uncertainty
relation, and set the minimal time at which a quasiprobability can become
non-positive, which is evidence for the onset of non-classical traits in the
system dynamics. As an illustrative example, we apply these results to a
conditional quantum gate, by determining the optimal condition giving rise to
non-classicality at maximum speed. Our analysis also hints at boosted power
extraction in genuinely non-classical dynamics.
