Uncertainty of the past

Certainly, a quantum wave may evolve backwards in time as well as forwards. The problem is that this sets up a causal relationship in which the future state is the cause and the past event the effect.

But this conflicts with the thermodynamic evolution of the Universe, which is to say the steady increase in entropy of systems, in which effects are always seen to follow their causes in time.

What we might call the probable set of Big Bang singularities which a theorist would consider viable, is probabilistic only due to our ignorance, it is a classical probability. We know that quantum probabilities obey different rules, and have no reason to apply them to the idea. If the end of time is a singularity, then our understanding of it may be said to comprise both forms of probability - a classical ignorance of the quantum probabilities.

I would say, that there is no probabilistic past, because of causality. Already in special relativity causality is a huge constrain. In quantum mechanics Lorentz invariance and time reversal symmetry is that could make you think that the uncertainty is possible backwards. But you have to consider one thing: the real world is not well approximated by "simple models". In quantum mechanics the environment of a given particle system will define the "arrow of time" locally. You are an observer with limited knowledge (you know only your past, but not the entire past hystory of everything. )

If there exists a wavefunction of the whole universe then it will obey thermodynamics. Let's continue the discussion with the assumption of the existence of such a wavefunction.

Regarding the big bang : lets work with the singularity model (as yin your question) , although it's quite an outdated view. The singular big bang is a very low entropy state, because it describes a fully symmetric isotropic homogenous spacetime. If the universe is closed and there is the slightest perturbation of this homogeneous system, it will immediately define a set of changes that will lead to an increased entropy state uniclverse. If there is a corresponding lagrangian, you can derive the future of your desired object (particle, observer... Whatever) as it will be forced to follow the clasdical solution that minimizes the action. The courses of actions on the macroscopic scale looks deterministic although quantum fluctuations can perturbe the system to deviate from it (so the future is not deterministic according to the coppenhagen interpretation)

But the arrow of time is defined by the classical solution, and a causal set of events will follow each other in a caussl way for every observer. If you have the knowledge about the universe (let's say you know the whole wavefunction of the minisuperspace), then you can derive all the allowed futures (including the perturbations) starting from the big bang. The events will select only one from the many (but finite set).

Backwords it cannot work, because the previous event has lower entropy and you have a loss of information in some sense. Imagine it through this analogue: if you go towards a higher entropy state, you differentiate, if you want to go backwards in time you integrate. During integration you will have an extra variable, an extra unknown parameter that could be anything. It's ambiguous, because there are no constraints ! (it's a bit vague, but contemplate on it!)