In QM, the wave function (in the Copenhagen interpretation) is not an actual physical wave but a device to derive probabilities about the outcomes of experiments. The wave function encodes all the information about the system we want to derive predictions for. Predictions are about future measurements. Once the measurement has been performed and the result is known, we adjust accordingly our expectation: the so-called collapse of the wavefunction just took place (let me add, in our minds). This subjective knowledge about the predictions of QM is crucial to avoid problem with causality in relativity when studying entangled systems. Fine.
What I am a bit confused about is what QM says about the past, rather than the future. What is the analog picture that QM gives about the state of a system in the past? What does QM say about the conditional probabilities of events? What does QM tell about, say, cosmology and the far past of the universe when e.g. string theory becomes relevant? I hope it is not a trivial, naive, question.