Abstract: We consider a quantum system that is being continuously monitored, giving rise to a measurement signal. From such a stream of data, information needs to be inferred about the underlying system's dynamics. Here we focus on hypothesis testing problems and put forward the usage of sequential strategies where the signal is analyzed in real time, allowing the experiment to be concluded as soon as the underlying hypothesis can be identified with a certified prescribed success probability. We analyze the performance of sequential tests by studying the stopping-time behavior, showing a considerable advantage over currently-used strategies based on a fixed predetermined measurement time.