Could gravitational waves have a large entropy in quantum gravity? The question is broad, and we still don't know what is quantum gravity. But let me be more specific. 
In the question (on this site, see below) on whether gravitational waves have entropy the answers were that it could, but typically a small quantity, or perhaps none at all when a solution of the Einstein equations. Much thanks for the answers to Peter Diehr, Lawrence Crowell, and wetsavannaanimal-aka-rod-vance. 
See: Do gravitational waves have entropy?
However, the discussion was in the realm of classical General Relativity (GR), with no quantum effects. The question excluded Hawking radiation type effects also. The question here is whether the results are different if we take quantum gravity into account (not just quantum fields in classical gravity)
In classical GR it was thought that black holes have no entropy, or very little, because of the no-hair theorem. Gravitational waves, classically, which interact very weakly with matter (like any gravitational field), would thermalize very slowly and thus acquire little randomness to have much entropy (using a Shannon or statistical mechanics measure of entropy, or so it seems on intuitive/physical grounds). The degrees of freedom that the gravitational waves could statistically have, say for the gravitational radiation emitted by the merging black holes (BH's), and detected in 2015 by LIGO, is not obvious, but classically it does not look like much. The waves seem to be a pretty well defined (even if we don't calculate to all orders) by the two black holes parameters. See Ref 1 for discussion and answers which explain it much better than I could, and which seemed to me to be approximately correct.   
The question is whether the same would be true taking quantum effect into account. This question was suggested in that other question, by wetsavannaanimal-aka-rod-vance. 
Even though BH thermodynamics (ie, end entropy >= initial entropy, and the BH entropy/mass/area equations) hold up consistently (from the observed data and deductions), it is interesting that the 3 solar masses, which when they were part of the BH contributed to a maximum measure of entropy, after radiation seem to be contributing much less. Of course, the total entropy is still more, so no physics was violated. Possibly some amount of entropy could be involved in the angular momentum of the gravitational waves (some must have been radiated away), but again that seems to have been a pretty deterministic process, not clear where a lot of randomness could come from.
So the question is, could the entropy be much different when taking quantum gravity into account?
One possible option could be whether there is anything that could be concluded from the AdS/QFT correspondence and the holographic conjecture, or a calculation from it? -- this was suggested clearly bywetsavannaanimal-aka-rod-vance in his comments, but the wording (and any misunderstanding) is mine 
Or would somehow the strong gravitational field that created the gravitational waves during the merger (all 3 phases, mainly in the strong field region) really need to include its effects on quantum gravity fields (but we're not at the Planckian scale yet, so probably not)? 
If that gravitational radiation was somehow focused some and a large part absorbed by a larger BH (so that it has a significantly smaller lambda than the size of the larger BH and the cross section is higher (there are papers on absorbing grav radiation by a BH, very lambda and geometry dependent, also spin, but even no spin will absorb), the BH entropy (and area) would have to grow because of the mass-energy of the gravitational wave absorbed. That certainly provides a max to the entropy the gravitational wave can carry (easy, same as a BH with its mass), but it does not provide a minimum. 
Is there anything one could conclude or suggest with our current understanding of quantum gravity and/or the overall physics? 
 A: Nice question.
Gravitational waves are solutions of the linearized equations of motion and as such I don't expect quantum gravity effect to change in a significative amount the entropy content of the waves. 
Nevertheless there are quantum gravity approaches, for instance the fuzzball proposal in string theory, where the discrepancy with classical physics is horizon-sized, so tiny differences in gravitational waves emission are expected. Indeed every fuzzball is different from another (fuzzball have microstates, while a black hole in general relativity is a single object with zero entropy from the microscopic point of view).
At present there is still no realistic model in 4d (but people are working on this) and studying dynamical process is very diffcult, so it's hard to give quantitative statements. 
What is radically different for sure is the way the information exit from the evaporated black hole, but this happens on a way longer timescale. In General relativity there is no way to do this, and the information paradox arises.
