Is the number of possible values of a quantum observable finite, countably infinite or uncountably infinite? Do we know, or have a theory about, whether the number of possible values any fundamental quantum property can assume upon observation is finite, countably infinite or uncountably infinite?
 A: We have a credible reason to think that the number of distinct possible results from any given observation is necessarily finite — even in principle, not just in practice.
The reason comes from thinking about what is required in order to reconcile quantum physics with gravity in a universe that is asymptotically like de Sitter spacetime, which appears to be the case in the real world. In such a spacetime, any given observer can only "see"
(by any means whatosever) a finite portion of the universe, much like an observer outside a black hole cannot see the inside. Any such event horizon has an associated entropy, which is roughly interpreted as the (natural log of the) number of mutually orthogonal states of the unobservable part that are consistent with the observable part. The key is that this entropy is finite, according to the famous Bekenstein-Hawking entropy formula — here applied to the cosmological event horizon instead of to the event horizon of a black hole. If the number of mutually orthogonal possible states in the observable part of the universe is finite, then any given observation cannot have more than a finite number of distinct possible outcomes, not even in principle.
Here's an excerpt from reference 1 that relates the finite entropy of de Sitter spacetime to the finite entropy of black holes:

Imagine an observer inside DS [de Sitter] space trying to contradict our contention by collecting as much entropy as she can. As long as she works on scales smaller than the DS radius of curvature, she can do this most efficiently by forming flat space black holes, whose entropy is bounded by their area. The black hole size is bounded by something of order the horizon size so there is no way to violate our bound. Put another way, a system with an entropy larger than the DS horizon size would simply not evolve into an AsDS [asymptotically de Sitter] spacetime...

That paper has been cited more than $300$ times, including by reference 2, which says this:

The absence of a classical de Sitter limit [in the best-understood theories of quantum gravity] suggests that the possible values of $N$ [the exponential of the entropy]... are sporadic, rather than arising from infinite families, and that there might be only finitely many choices [all of which are finite].

That paper has been cited more than $600$ times. This is all still a work in progress, but the authors of these papers are well-respected experts in quantum gravity research, so I suppose this qualifies as mainstream physics. Reference 3, also written by one of the most well-respected experts, is a more recent example of the ongoing research in this area.

References:

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*Banks, Cosmological Breaking of Supersymmetry? (https://arxiv.org/abs/hep-th/0007146)


*Witten, Quantum Gravity In De Sitter Space (https://arxiv.org/abs/hep-th/0106109)


*Susskind, Black Holes Hint Towards De Sitter-Matrix Theory (https://arxiv.org/abs/2109.01322)
