Gary.
Okay, I'll try to answer: "I understand that QM and GR are both 4-D models. Presumably, they both share the same three spatial dimensions.
What is the basis for the belief that they also share the same 4'th dimension?"
QM is not a 4 dimension model.
Let's look at in the context of Bell's theorem. Even though the theorem claims to test local realism, the choice function is invested in detector settings, an unrealistic proposal. After repeating random choices a statistically significant number of times, we count the number of times the choices correlate and normalize the results. Richard Gill says, "The physicist's correlation is just the probability of equal outcomes minus the probability of different outcomes."
Quantum theory confuses physics with physicist. Random choices yield random results. And the time parameter is non-existent. In other words, the theorem proves just what the theorem assumes (See Karl Hess, Einstein was Right! ).
In an objective spacetime, simple connectedness is a theorem by existence -- the added degree of freedom that the time parameter provides assures objective outcomes without pinning an observation to a particular observer. It fulfills Einstein's requirement to complete an observation without disturbing the system. In other words, quantum mechanics can be derived from continuous spacetime in an objective manner, though the converse is not true. Quantum theorists make all kinds of ad hoc assumptions -- Quantum Bayesian is the worst of them, as it imposes a measure of personal belief.
The time coordinate has a negative value in Minkowski spacetime. Time reversibility is an absolute requirement of general relativity -- it provides symmetry -- while special relativity, straight line motion, is asymmetric with regard to time. This is how quantum mechanics can claim compatibility with special relativity, and why QM and general relativity cannot both be foundational.
Special relativity requires time dilation, something QM ignores. Or rather, normalizes, and then is able to ignore. Since time is length 1 in one direction, it has to be length 1 in all directions. Fine. In what measure space? Quantum theorists are in a potential trap of logical contradiction: if the measure space is n-dimension Hilbert space, it has to be rigged for 3 dimensions, R^3. So it limits itself to a 3 coordinate system, xyz, though we know that a complete accounting for position requires a 4 coordinate system, xyzt. Because normalized time zeros out of an algebraic equation, continuity can never be built into a quantum theory; i.e., T = 0 in a linear equation.