According to the Everett many-worlds interpreration of quantum mechanics the huge dimensionality of the many-particle wave function implies existence of parallel universes in which all possibilities occur simultaneously. While this interpretation might appear to be at odds with Occam's razor it has far fewer postulates than its competitors. It is also true that the information related to alternative histories in some form or another is currently required to predict experimental observations.
Time evolution of the wave function in general is always deterministic. Thus when we ask questions such as why we experience one future instead of another the typical answer is that there is nothing special about any particular world since all possible worlds exist and we ask the same question in all of them (at least in all relevant to this question). Quantum jumps are a consequence of short timescale decoherence, but decoherence itself is also governed by deterministic evolution of the wave function.
The experience at least in some sense is deterministic even for the consciousness, it's just that something violent is continuously happening without its knowledge, but if the consciousness knew everything there is to know about the world, it could predict what's going to happen to them as much as a split brain patient or anyone about to experience a qualia they've never experienced before can ever be expected to be able to understand what's in store for them (ignoring the fact that they can't know everything there is to know about the world).
The practical expectation values are among other things a result of the number of universes in which particular outcomes happen. One has to calculate the sum of all possible wave functions (from all the universes) to find a priori probability distributions, but that's another story. Anyway, such a theory allows one to make physically meaningful predictions which are accurate and highly consistent with observations.
However, while for an objective observer there is no particular problem as one branch of the multiverse is as real as any other, an individual conscious observer will likely be quite unsatisfied by the lack of practical deterministic predictions, but even if there was no problem from the point of view of the physicist, for a philosopher the hard problem of consciousness remains.
The hard problem of consciousness is basically asking what if any is the difference between a person who experiences consciousness and a system which simply reacts identically to a conscious person, but doesn't truly experience anything? Not everyone believes there is any problem associated with the question, but one has to consider that according to general relativity time isn't unique or fundamental in a way that would appear necessary for the experience of present, i.e. there is no passage of time and no unique present exists, rather time may simply be a kind of dimension no different from space. How then could consciousness ever experience the passage of time as we appear to experience it? The question becomes what sets apart the present moment from the past and the future for the consciousness when physically there doesn't appear to be anything that could do so?
It has been suggested that time is an entanglement phenomenon and arises from quantum mechanics. Such a time is meaningful only in relation to some other quantum variables often considered to be of thermodynamical origin. Thermodynamics itself appears nothing more than unlikely (low entropy) microstates evolving (chaotically) into likely (high entropy) microstates (that greatly outnumber low entropy ones). The standard problem of time is asking why the universe appears to have started in a highly unlikely low entropy state (perhaps anthropic), but that's separate from the problem of time in relation to consciousness. In a static set time evolution is simply a direction. How could this give rise to a meaningful present moment necessary for consciousness is at best unclear.