Standard formulations of quantum mechanics feature two distinct laws describing how the state of a quantum system changes in time. The first law, the Schrödinger equation, is smooth and time-reversal symmetric. By looking at a state evolving according to this equation one cannot tell in which direction time is flowing. The second law, the collapse postulate, is abrupt and time-reversal invariant. It says that, upon measuring a property of the system, its state suddenly changes to a new one, reflecting the outcome of the measurement. Because the state of the system depends on the results of past measurements, it is commonly believed that quantum mechanics says that physics is time-orientated. There seems to be a quantum arrow of time.
However, quantum theory also predicts that the quantum state can never be observed. We shoe that when sticking to what can be observed, the predictions of quantum theory are entirely time-reversal symmetric. Quantum theory allows to calculate transition probabilities: answers to the question “what is the probability of observing A at time t1 given that B is observed at time t2?”. And these probabilities are independent of the direction of time, they do not change t1 is before t2 or the other way around. This means that not only the quantum state can never be observed, its time orientation has no observable consequence.
Some interpretations of quantum mechanics hold that the quantum state is not physical, but just a mathematical tool to help in the computation of probabilities. In this way we can understand its time-orientation: the state reflects past measurements because we humans know the past but not the future, and thus we assign states to systems that reflect past observations.