  Controlling dynamical fluctuations in open quantum systems is essential both
for our comprehension of quantum nonequilibrium behaviour and for its possible
application in near-term quantum technologies. However, understanding these
fluctuations is extremely challenging due, to a large extent, to a lack of
efficient important sampling methods for quantum systems. Here, we devise a
unified framework --based on population-dynamics methods-- for the evaluation
of the full probability distribution of generic time-integrated observables in
Markovian quantum jump processes. These include quantities carrying information
about genuine quantum features, such as quantum superposition or entanglement,
not accessible with existing numerical techniques. The algorithm we propose
provides dynamical free-energy and entropy functionals which, akin to their
equilibrium counterpart, permit to unveil intriguing phase-transition behaviour
in quantum trajectories. We discuss some applications and further disclose
coexistence and hysteresis, between a highly entangled phase and a low
entangled one, in large fluctuations of a strongly interacting few-body system.
