Quantum dynamical systems have attracted increasing attention in both academia and industry over the past decade. Owing to intrinsically quantum phenomena such as superposition, entanglement, and measurement-induced state change, their behaviors are considerably richer than those of classical dynamical systems. In particular, because quantum states cannot be accessed without measurement, monitoring temporal behaviors of quantum systems requires a treatment fundamentally different from the classical one. In this paper, we investigate the monitoring of quantum dynamical systems in \emph{continuous time}. We model the observable behavior of such a system through a \emph{discretized observational model} obtained from a sequence of quantum measurements, which induces a branching process. To specify temporal properties, we employ signal temporal logic for reachability and extend it with value-freezing operators to express richer patterns such as oscillation and peaking. Based on this combination of branching-process semantics and temporal logic, we develop monitoring algorithms that decide whether a model satisfies a given formula. The algorithms run in time polynomial in the size of the model and linear in the size of the formula. Experimental results on case studies and benchmark instances demonstrate the effectiveness and scalability of the proposed method.