What does QM say about the past rather than the future?

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)