What does QM say about the past rather than the future? 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.
 A: Quantum mechanics can be used to answer questions about the past in a fairly straightforward way as any question of that type can be phrased as a question about expectation value of operators (or as transition amplitudes).  As a simple example consider a two state system (e.g. spin 1/2).  Suppose someone else prepares the state in either spin up or spin down but doesn't tell you.   Also suppose that the dynamics are unitary and known ($U$).  Then you can use quantum mechanics to ask, for example, what is the probability that the state was prepared in the 'up' state if I measure it in the up state now?
$$p = | \langle \mbox{up}|U|\mbox{up} \rangle|^2$$
So really there is nothing new, you just apply quantum mechanics to whatever question you mean to ask about the past. You might have to be a bit careful in phrasing the question however.
For the general case of reconstructing the past state given present measurements, see for example the wikipedia article on Quantum tomography (http://en.wikipedia.org/wiki/Quantum_tomography)
A: On the fundamental level, QM is time-symmetric, hence it says the same about the past as about the future. The dynamics of the state is deterministic and given by the quantum Liouville equation (or, if you consider an isolated system in a pure state, by the Schroedinger equation). This state determines the probability distribution of measurable events at any time, the past as well as the future. 
Accounting for new information about the past (or the future), as it becomes used by the observer to improve predictions about the past (or the future), is accounted for by projecting the state to the invariant subspace determined by the new information. This is the quantum analogue of taking conditional expectations in a classical stochastic model when new information about the past  (or the future) becomes available. 
