Revivals of Bell nonlocality require Schr\"odinger and Heisenberg non-Markovianity

Bell nonlocality is a key resource in quantum information, demonstrating the nonclassicality of quantum theory. Noise, however, is in general detrimental to nonlocality, and can cause the loss of the ability to violate any Bell inequality. Memory effects, on the other hand, can restore this quantumness and, as recently shown, they can be differently characterized in the Schr"odinger and in the Heisenberg picture. Here, we show that if memory effects allow for revivals in time of nonlocality, then the dynamics must be non-Markovian in both pictures. We showcase our findings through a device-independent quantum key distribution task, for which Bell nonlocality is necessary.