In a classic paper, Bohr and Rosenfeld (1933) showed that the quantization of matter implies the quantization of the electromagnetic field. In other words, it is logically inconsistent to have classical electromagnetic waves interacting with a quantum measuring apparatus. Crucially, the same is not true for gravity, due to the existence of a fundamental length scale, the Planck length.
In fact, not only is the quantization of gravity not a logical necessity, it is also unobservable within Einstein gravity. Freeman Dyson first showed that it is impossible to detect a single graviton with high probability in any realistic experiment, and conjectured a censorship effect that precludes the observation of the quantization of the gravitational field in Einstein’s theory. Dyson's argument is based on the striking fact that the absorption cross section for gravitons is universal and equal to the planck area within numerical factors of order one.
Dyson’s work was motivated by the hope that the failures in reconciling General Relativity with Quantum Mechanics were really due to the fact that the gravitational field is a purely classical entity. In the dichotomous world he envisioned, the geometric theory of gravity would peacefully coexist with the quantum realm, and the obstruction in observing individual gravitons would be attributable to their non existence.
In our own paper, we propose a type of experiment that would be able to distinguish between classical and quantum gravity at the observational level. The experiment consists in looking for dark lines in gravitational spectra, due to gravitational absorption by quantum bound states. Gravitational absorption lines would probe quantum gravity in two distinct ways. First of all, since General Relativity forbids them, they would explore and constrain exotic physics close to the Planck scale. Secondly, and perhaps more importantly, they would confirm that the gravitational field is quantized at low energies, effectively proving the existence of gravitons.