Causal Dynamical Triangulation
What it (like LQG) has to recommend it is that the structure of space emerges from the theory itself. Basically, it proposes a topological substrate (spin-foam) made of simplexes (lines, triangles, tetrahedrons, etc). Spatial curvature emerges from how those simplexes can join together.
Degeneration and the arrow of time
The key insight for CDT was that space emerges correctly if edges of simplexes can only be joined when their arrows of time are pointing in the same direction.
So time doesn't emerge?
To say the say thing in a different way, the rule isn't that the arrow of time makes entropy increase, it's that when you have an entropy gradient along a time-like curve, you have an arrow of time.
The appeal is that we don't have to say that the time dimension has special rules such as making entropy increase in one direction. Also, both QM and relativity show us a time-symmetrical picture of fundamental interactions and emergent arrow-of-time doesn't mess that picture up.
Observables and CDT
In quantum mechanics, you can only observe certain aspects of a wavefunction, called Observables. Given a superposition of a arrow-matched and arrow-mismatched CDT states, is it the case that only the arrow-matched state is observable? Ie that any self-adjoint operator must be only a function of arrow-matched states?
I frankly don't know CDT remotely well enough to say, but it doesn't sound promising and I have to suspect that Loll et al already looked at that.
A weaker variant
But as far as my limited understanding CDT goes, with all due humility, there's room for them to be mostly censored. Like, arrow-mismatched components are strongly suppressed in all observables in cases where there's a strong arrow of time.
Degeneration: A feature, not a bug?
Suppose for a moment that CDT is true but that the "only join edges if arrows of time are the same" behavior is just emergent, not fundamental. What happens in the far future, the heat death of the universe, when entropy has basically maxxed out?
Space degenerates. It doesn't even resemble our space. It's either an infinite-dimensioned complete graph or a 1-dimensioned line.
The Boltzmann Brain paradox
What's the likelihood that a brain (and mind) just like yours would arise from random quantum fluctuations in empty space? Say, in a section of interstellar space a million cubic miles in volume which we observe for one minute?
Very small. Very, very small. But it's not zero. Nor does it even approach zero as the universe ages and gets less dense, at least not if the cosmological constant is non-zero. The probability has a lower limit.
Well, multiplying an infinite span of time times that gives an infinite number of expected cases of Boltzmann Brains exactly like our own. The situation should be utterly dominated by those cases. But that's the opposite of what we see.
Degeneracy to the rescue
Having described that degeneration, I can't help noticing its resemblance to the Big Rip, the hypothesized future event when cosmological expansion dominates the universe and tears everything apart.
That makes me wonder if the accelerating expansion of space that we see could be explained along similar lines. Like, the emergent arrow-of-time-matching isn't quite 100% perfect, and when it "misses", space expands a little.
This would fit with the weaker variant proposed above.
Edit: dynamic -> dynamical