  We consider quantum jump trajectories of Markovian open quantum systems
subject to stochastic in time resets of their state to an initial
configuration. The reset events provide a partitioning of quantum trajectories
into consecutive time intervals, defining sequences of random variables from
the values of a trajectory observable within each of the intervals. For
observables related to functions of the quantum state, we show that the
probability of certain orderings in the sequences obeys a universal law. This
law does not depend on the chosen observable and, in case of Poissonian reset
processes, not even on the details of the dynamics. When considering (discrete)
observables associated with the counting of quantum jumps, the probabilities in
general lose their universal character. Universality is only recovered in cases
when the probability of observing equal outcomes in a same sequence is
vanishingly small, which we can achieve in a weak reset rate limit. Our results
extend previous findings on classical stochastic processes [N.~R.~Smith et al.,
EPL {\bf 142}, 51002 (2023)] to the quantum domain and to state-dependent reset
processes, shedding light on relevant aspects for the emergence of universal
probability laws.
