In quantum mechanics you can let go of the requirement to test and measure, use instruments, have interaction with the environment, and make observations.
This applies to closed quantum systems.
In such a case, Newotnican mechanics and quantum mechanics share a very important foundational principle which says there always exists a unique future and unique past state.
In this way, quantum mechanics is as deterministic as Newtoinan yet with the difference that the sum of probabilities of all possible events has to add up to unity. When the future is unique you are absolutely certain that the sum of probabilities is always unit.
When the quantum state is closed there is one to one correspondence between the past and the future because of the unitary operator that defines the evolution of the quantum system both forward and backward.
Now consider some outlier and speculative theories, of course so far with no experimental evidence at all, that the future can indeed influence the past. This is called retrocausality.
It may also break down the foundational principle of unitarity.
Exploration of the mathematical implication of this violation is engaging at least in theory.
A few of them follow below but I wonder which one is logically incorrect or impossible?
1/ Without unitarity you will not have conservation of probability and will have loss of information. When probabilities do not add up to unity that will mean there exists events that are outside of the range of possible events. Also, if you lose information about the past in principle, perhaps you can say that the past is undefined and does not make sense at all. All that matters or is meaningful is the moment of the present and the future! Evolution is not reversible.
2/ Without unitarity, the map between past and future is not injective, so you can have multiple past states for each future state. All those pasts can be projected to the same future! This could be explained by the existence of a global (as opposed to a local) hidden variable either in an abstract or physical space, i.e. extra dimension.
3/All events in such a universe are intrinsically connected no matter how much they are separated or they share some unknown universal deeper property but inaccessible or unmeasurable to local observers forever.
4/ If the map is not injective, the kernel set is not zero. This means there might be many past states for which there is absolutely no future at all. Some past states actually evolve to nothing, they vanish in the future!
5/ Losing unitarity means losing the predictability of probabilities. Absolute unpredictability is indistinguishable from total chaos. What remains will be less foundational principles and more inconsistencies.
6/ When projecting the fiber of the past into the future, with the evolution not reversible, in some exotic topology related to some patches of the universe, the past simply vanishes and the future is just created every moment in the present without any past.