  We consider a class of open quantum many-body systems that evolves in a
Markovian fashion, the dynamical generator being in GKS-Lindblad form. Here,
the Hamiltonian contribution is characterized by an all-to-all coupling, and
the dissipation features local transitions that depend on collective,
operator-valued rates, encoding average properties of the system. These types
of generators can be formally obtained by generalizing, to the quantum realm,
classical (mean-field) stochastic Markov dynamics, with state-dependent
transitions. Focusing on the dynamics emerging in the limit of infinitely large
systems, we build on the exactness of the mean-field equations for the dynamics
of average operators. In this framework, we derive the dynamics of quantum
fluctuation operators, that can be used in turn to understand the fate of
quantum correlations in the system. We apply our results to quantum generalized
Hopfield associative memories, showing that, asymptotically and at the
mesoscopic scale only a very weak amount of quantum correlations, in the form
of quantum discord, emerges beyond classical correlations.
