There is no general connection between the quantum speed limit and non-Markovianity. (arXiv:1907.05923v1 [quant-ph])

The quantum speed limit sets a bound on the minimum time required for a
quantum system to evolve between two states. For open quantum systems this
quantity depends on the dynamical map describing the time evolution in presence
of the environment, on the evolution time {\tau} , and on the initial state of
the system. We consider a general single qubit open dynamics and show that
there is no simple relationship between memory effects and the tightness of the
quantum speed limit bound. We prove that only for specific classes of dynamical
evolutions and initial states, there exists a link between non-Markovianity and
the quantum speed limit. Our results shed light on the connection between
information back-flow between system and environment and the speed of quantum

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