It was my understanding that one of the hardest problems in the unification of GR and QM is "the problem of time". The axiomatic structure of quantum theory needs a "time axis" ; it doesn't necessarily need a "space axis", but there is a fundamental separation between "time" and "state of the system". The state of the system is described by a ray in Hilbert space, which is supposed to correspond to an "instantaneous" picture of the system, while time is fundamentally a parameter over which the unitary evolution is running.
Whatever are the degrees of freedom in the hilbert space, they can, or cannot, be "space-related" - that's not the difficult part. For instance, we could have some "spin" degrees of freedom which have nothing to do with space ; or we could have particle positions in an Euclidean space (in which case the degrees of freedom are those particle positions Xi,Yi, Zi - promoted to operators over Hilbert space) ; or we could have fields parametrized over 3 (or more) space coordinates x,y,z), this time, the FIELDS become the operators phi(x,y,z), parametrized in (real) x, y and z ; or whatever other exotic set of degrees of freedom you want to consider.
But time is set apart.
This is a priori a very "Newtonian" viewpoint: an "instantaneous" state, and a time evolution. In a Minkowski space, we can still save the day: we can have the time parameter correspond to the parametrisation of space-like planes, and somehow put constraints upon the different states and unitary evolutions, so that there are possible links between DIFFERENT ways of slicing up Minkowski space, so that they are in agreement (this is in fact requiring Lorentz-invariance of this unitary evolution). We can even go further: we can even save the day in a GIVEN CURVED SPACETIME. As long as the spacetime is GIVEN, we can do quantum theory over it, and require that the unitary evolutions, as seen by different "slicings" (and hence different choices of time coordinates) are in agreement.
However, there is a fundamental hick when one wants to introduce the DYNAMICS of spacetime (as is done in general relativity) ; because you bite your tail: in order to set up the quantum theory, you already NEED your spacetime to have your "time parameter" and your "degrees of freedom", while this time parameter is supposed to COME OUT of the dynamics, but it can't come out as long as you don't HAVE your time axis defined (over which the unitary dynamics is to be parametrized).
So the fundamental difficulty comes from the fact that this time parameter enters directly in the fundamental axioms of quantum theory, and is set up totally different from the other degrees of freedom (time = real parameter, other degrees of freedom = structure of Hilbert space). Some people close their eyes and think of England, and START with a given (flat?) spacetime, and define a tensor field over it that becomes ultimately the gravitational curvature ; these approaches are called "background dependent" and are against the spirit of GR ; superstrings suffers from it as far as I understand.
