How to quantify the "degree of quantumness" vs. "degree of classicality" of a physical system? Some physical systems are analyzed using QM or QFT, whereas others are analyzed using classical physics.
Question: How can one refine this qualitative distinction to yield some kind of quantitative measure of the "degree of quantumness" or conversely the "degree of classicality" of a physical system?
Notes:

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*Part of the issue in formulating this question is that I'm not quite sure what I mean by "physical system". Probably there are physical systems which are "more quantum" in certain states than in others -- so that "quantumnness" is not a property of the system, but of the data [system + a state of they system]. But perhaps there are also some physical systems which have more "capacity for quantumness" or "capacity for classicality" than others, i.e. maybe there are systems $X, Y$ such that the "maximal (or minimal) quantumnness" attainable by a state of $X$ is bigger than the "maximal (or minimal) quantumnness" attainable by a state of $Y$, in which case we might say that the system $X$ is "more quantum" than the system $Y$.


*One thing I'm pretty sure of is that if I have a system $X$ which I can conceptually divide into two subsystems $X_1$ and $X_2$, then for any state $S$ of $X$, there are various quantitative measures of how entangled $S$ is with respect to the division $X = X_1 + X_2$. It would seem reasonable to interpret this as a measure of how "quantum" the state $S$ is with respect to the division into $X_1$ and $X_2$. Perhaps one measure of "system classicality" of $X$ would be some sort of measure which minimizes the entanglement found across different subsystem-decompositions of $X$. Or perhaps there is some other approach along different lines.
 A: I don't think that I'm really going to answer your question, but perhaps this will get the ball rolling - so to speak.  Your comment about "capacity for quantumness" and "capacity for classicality" got me thinking of something I read just the other day.
In Eisberg and Resnick's Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, the chapter on Quantum Statistics, they write:

[The Boltzmann distribution] can be applied in all quantum systems at energies more than serveral kT above the ground state - the states are sparsely occupied so that n(Boltz) [value of Boltzmann function] is very much less than one.

So, though they do not use the words you use, (I've never seen anyone use those before), they  indicate that measuring the energy above ground state could be used as a parameter on whether you need to use BE, or FD statistics (quantum regime), or you can use Boltzmann statistics (classical regime).
As an example, they write:

The  Maxwell distribution of speeds of gas molecules moving freely inside a box is validly deduced from the Boltzmann distribution because n(Boltz) for all free particle states is very small under the conditions usually existing in nature for ordinary gases.

As the temperature is lowered and/or the density of particles increases, then their quantum nature would have to be taken into account and the correct statistics used.
As a further example, it is always noted that electrons in a metal can, to a first approximation, be modeled as a gas, but one in which the density of electrons means that we must treat them with FD statistics.  In the language of QM, electrons in a metal are a degenerate system.
A: This isn't really an answerable question, but here is how I'd frame it. People have, since Planck, computed various quantities classically and with the benefit of QM, compared the two, and called the $O(\hbar)$ difference a "quantum effect".
However, you can't compute the stability of matter classically, not the detailed properties of most materials, which are all undergirded by elaborate concatenations of quantum effects. Our world is deeply quantum. So your question quickly devolves to

When can I ignore QM interference effects and effectively  describe situations classically, modeling them by classical theories?

This is a comparably complicated, mind-bending, question, and in each field different seat-of-the-pants popular criteria have emerged. In formulations of QM in phase space, lending themselves to superficially "natural" comparisons with classical mechanics,
a popular criterion is whether the Wigner function, the quasi-probability density in phase space has negative regions, a sufficient but not necessary hallmark of quantum behavior. Such regions, hiding behind the uncertainty principle of $O(\hbar)$, could be arguably quantified, but that is not without its own controversies.
Nifty computer investigations illustrate such effects. (Without numerics, you might, or might not, get something out of my own video.)
There is a technical, information theoretical, technique in quantifying
"quantumness" through entropy, (see a talk and references therein), which monitors the loss of information implicit in taking the classical limit. However, it relies on the crucial "quantum offset" of the normalization of the phase-space volume element, which many have unfortunate conceptual problems with, so they shrug it off before they get to the bottom of it. It can frame the picture  in  the  simplest systems, but it is notoriously hard to calculate more complicated systems with.
