Would the presence of B-modes in the CMB provide evidence for quantum gravity? Finding B-modes in the CMB (which aren't due to foreground contamination) would be evidence for gravitational waves, because they cannot be produced by density perturbations (to first order, is my understanding). My question is how can we jump from "We have found gravitational waves in the CMB" to "this is evidence of the quantum nature of gravity"? (this is a common enough statement in science publications on the topic) According to this answer, the size of the universe before inflation was ~500k Planck Lengths, and after inflation it was the size of a grain of sand. So during the entire process, the scales involved were many times the scale we usually associate with quantum gravity. It seems to me that finding gravitational waves in the CMB would give more evidence for the validity of classical GR, not of quantum gravity.
Just to clarify, this question is not "do the B modes found by Planck suggest an observation of gravitational waves" (since that is under debate) or "does the gravitational field have a quantum description?". I am just interested in how one connects the scales associated with inflation to the certainty that observation of gravitational waves would probe the quantum nature of gravity.
 A: The question whether primordial B-mode detection says something about quantum gravity or not may be subjective, but one can give an unambiguous and objective answer why people were even bringing up quantum gravity in connection with B-modes (and BICEP2). Here, I will try to answer that question.
My answer is mostly based on an excellent (guest) blog post by Liam McAllister, a researcher from Cornell who has published several papers on inflation, and is currently writing a textbook called String theory and inflation together with Daniel Baumann from Cambridge (part of the draft is freely available here).
In the post, McAllister explains that a generic prediction of inflationary models is that the tensor-to-scalar ratio $r$ of the primordial gravitational wave perturbation spectrum is related to the energy density $\rho$ at the time during which inflation occurred: 
$$\rho^{1/4}=2.2\times 10^{18}\;\text{GeV}\left(\frac{r}{0.2}\right)^{1/4} $$
Thus, if a detection like BICEP2 determines that $r$ is on the scale of $0.1$, this tells us that we can hope to gain insights into physics that occurred at energy scales that we could have never dreamed of achieving here on Earth (for comparison, the LHC runs at $\sim 10^{4}\;\text{GeV}$). In fact, this energy density is reasonably close to the Planck scale, which is where it is generally expected that quantum gravitational effects should start kicking in. Reminding ourselves that these gravitational waves are theorized to have originated from quantum mechanical fluctuations, we seem to be tantalizingly close to realizing the dream of experimentally accessing regimes where quantum gravity can be probed!
...this, of course, all under the assumption that BICEP2-like results have been demonstrated which, at the moment, seems dubitable at best.
A: This is rather subjective. But I'll volunteer my answer: 
Would the presence of B-modes in the CMB provide evidence for quantum gravity?
No. And recent argument is that they don't even provide evidence for gravitational waves. I don't know if you saw the recent Horizon program, but note that what didn't come out is that the gravitational waves date from "the instant of creation", whilst the CMB dates from 300,000 "maelstrom" years later. Think blender. Also note Paul Steinhardt features in the program. See Physicist Paul Steinhardt Slams Inflation, Cosmic Theory He Helped Conceive. Even inflation is looking shaky IMHO, like it's a solution to a problem that doesn't exist, and big bang cosmology is better off without it.  
How can we jump from we have found gravitational waves in the CMB" to this is evidence of the quantum nature of gravity?
We can't. We haven't even detected gravitational waves. What we have here is an inference atop a speculation riding on a unproven hypothesis, none of which address the E=hf quantum nature of an electromagnetic wave, which by the way has an active gravitational mass, and therefore is a gravitational wave of sorts too.  
