Difference between QFT In curved spacetime, semiclassical, and quantum gravity?

• QFT in curved spacetime: The spacetime metric is fixed. It doesn't react to the state of the quantum fields, which are described by operators on a Hilbert space. The fixed metric field defines what "spacelike" means, which is important because observables built from the quantum fields are required to commute with each other at spacelike separation. (This prevents faster-than-light communication.) This approximation is self-consistent. Hawking radiation can be derived using this approximation, but the black hole never evaporates in this approximation, because the spacetime metric does not react to what the quantum fields are doing. A special case of QFT in curved spacetime is QFT in flat spacetime, which of course is very well-tested. I'm not aware of any direct tests of the effect of spacetime curvature on quantum fields, although there are plenty of tests of the effect of spacetime curvature on classical approximations of quantum fields. Tests of quantum effects in free-fall don't count, because they're not sensitive to the curvature.

• Semiclassical gravity: This name often refers to an approximation that attempts to account for how the spacetime metric would react to the quantum fields, but still using a classical description for the metric (so that the metric commutes with all operators on the Hilbert space). The stress-energy tensor $T_{ab}$ is a non-commuting operator, so to describe the behavior of the metric field using something like Einstein's field equation, this approximation uses the expectation value of $T_{ab}$, like this: $R_{ab}-frac{1}{2}g_{ab}Rproptolangle T_{ab}rangle$. This approximation is not strictly self-consistent , but it's usable as long as we avoid situations involving quantum superpositions of very different values of $T_{ab}$. It has been used, for example, to try to account for the fact that a black hole should lose mass as it radiates. The inconsistency is explained in this blog post: Luboš Motl (2012), "Why 'semiclassical gravity' isn't self-consistent," https://motls.blogspot.com/2012/01/why-semiclassical-gravity-isnt-self.html

• Quantum gravity: This refers to a fully quantum theory with no fixed background metric (except for its asymptotic structure)$^*$, but which nevertheless exhibits gravity in a way that is well-approximated by general relativity under appropriate conditions. The correct theory of quantum gravity remains to be determined. Currently, the best-understood example of this kind of theory comes from the AdS/CFT correspondence. This can be regarded as defining a theory of quantum gravity (string theory) in asymptotically-AdS spacetime in terms of a conformal quantum field theory (CFT) associated with the asymptotic structure. CFT is relatively well-understood mathematically, but exactly how spacetime and gravity manage to emerge is still an active research area, as is the question of how to extend the idea to spacetimes with more realistic asymptotic conditions (namely de Sitter).

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$^*$ I don't know of any well-developed proposals for how to get by without a fixed asymptotic spacetime structure, and I have seen general arguments that such a structure is necessary. Of course, any general argument is only as good as the assumptions that go into it, and since we are on unfamiliar territory when dealing with quantum gravity (due to the lack of any prescribed geometry or even any prescribed causal structure in the bulk of spacetime), it is difficult to know how watertight those general arguments are.