There's no problem reconciling the quantum with the Newtonian. Quantum mechanics recovers Newtonian mechanics in the appropriate limit. The problem is reconciling the quantum and the Einsteinian.
Actually, Newtonian gravity can be added to QM and work perfectly well. It's GR gravity that doesn't work with QM, especially if you try to model very high curvature like you'd get near a black hole.
Quantum Electro-Dynamics (QED) is the application of Special Relativity (non-accelerating frames of reference, i.e. moving at a constant speed) to Electromagnetism. Thus, the issue is with applying accelerating frames of reference (the General in GR) to QM.
None of these have anything to do with what I said. SR works just as well as classical mechanics with acceleration. If SR didn't work with acceleration, it would have never been accepted as a valid theory at all, it would have been a laughing stock, as acceleration was well understood since the times of Newton.
What general relativity does different from special relativity is that it extends the concept of relativity from inertial frames of reference to all frames of reference, even accelerating ones. In the process, it also explains the reason why inertial mass and gravitational mass happen to be the same, by tying the gravitational interaction fundamentally to the concept of acceleration.
> Special Relativity (non-accelerating frames of reference, i.e. moving at a constant speed)
Sorry, but this is a pet peeve of mine: special relativity works perfectly well in accelerating frames of reference, as long as the spacetime remains flat (a Minkowski space[1]), for example when any curvature caused by gravity is small enough that you can ignore it.
