Quantum mechanics is a non-commutative probability theory. As such, it fundamentally behaves differently from classical probability theories. This manifests itself most pronouncedly in the uncertainty principle: a given state can not be assigned a definite position and a definite momentum at the same time.
Enter the measurement: If I understand correctly, when performing a measurement the outcome is a definite result on a classical level. I.e. once we have measured say the position of a particle, the information about where it was is saved in some classical way, where the classicality here emerges through having a large enough system.
To understand this apparent disparity, the concept of wave function collapse was regarded as the solution for a long time, e.g. as part of the Copenhagen interpretation of quantum mechanics. Nowadays it is widely accepted that there is no such thing as collapse, instead the quantum state of the universe evolves in a unitary way (i.e. by the Schrödinger equation). The apparent collapse is then explained as a result of interactions in many-particle systems (see e.g. on SE: this and this, and links therein. In particular this.). This also explains how some many-particle systems may well be approximated as classical and can store the information of measurement outcomes.
The question: Let us suppose we know the complete quantum state of the universe (or a closed system for that matter, to address StéfaneRollandin's concerns that the universe's quantum state may be ill-defined). Can we predict measurement outcomes in the future? Or can we only assign a classical probability? To reformulate: Is quantum mechanics on a measurement level a deterministic theory or a probability theory? If it is the latter how can this possibly be consistent with unitarity as described above? And are the probabilities associated still part of a non-commutative probability theory?
Note that in this question, interpretations of QM will not play a role, since by definition they yield the same theory in terms of observable quantities. I would therefore appreciate if an interpretation free answer could be given.