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I would be far more careful using quantum physics in informal "philosophical" arguments. In most instances, people summon quantum effects to create a feeling of answered question, while in fact the answer is confused or, worse, not an asnwer at all. The general rule is: every philosophical argument using the word quantum is bogus. (Take with a grain of salt, of course.)
More concretely, closed quantum systems (i.e. when no measurement is done) evolve deterministically, and their evolution can be periodic.
I thought that in closed quantum system there are only probabilities of a true indeterminisitc nature - and the only deterministic part is at the collapse of the wave function (where the positions, speed,... are truly determined - but impossible to measure correctly).
Still the fact remains that one universe is holding observers and even there is only one sollution to past eternity - that of a cyclic universe of the same kind and same parameters of the big bang - the futures of the universe would be determined by the acts of those observers. Different acts...