The one difficulty I see with LQG is that it requires an enormous number of degrees of freedom, e.g. these spin variables in the net. This is in contrast to stringy holographic theory where the fields in a space are equivalent to fields on a boundary or a horizon of one dimension lower. In this setting entropy of a black hole is the entanglement entropy of states interior and exterior to the black hole. This reduces the amount of data, and thus entropy, required.
Are there suggestions, conjectures or maybe serious theory which attempts to describe the spin variables of LQG according to such entanglements in string-brane theory?
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I would say that the it is the other way around. The number of degrees of freedom in LQG is not "enormous", and it is much less, and not much more that in strings. The spin variables on the net are just the same variables as in classical general relativity, namely the metric, except that there is a cut off at small scale. So, there are infinitely many less variables than in classical general relativity. because the degrees of freedom with wavelenght smaller than the Planck length exist in classical GR but not in LQG. It is true that in strings one might expect an holographic principle that reduces the degrees of freedom to the boundary theory, but this is not a boundary a theory in our physical spacetime. It is a boundary theory in a spacetime of higher dimensions. So, the boundary theory has still more dimensions than our space, and therefore the number of degrees of freedom is still much bigger that in LQG.
About the stage of development, certainly both loops and strings are very preliminary, and not fully understood, but I would definitely say that LQG is far better understood (it is a simpler theory than strings). In LQG we know the fundamental degrees of freedom and we can write the basic equations of the full theory in a compact form in a few equations. See for instance my recent review paper http://fr.arxiv.org/abs/1012.4707. In strings we do not yet know the fundamental degrees of freedom, and we only know certain "corners" of the theory, with many indications that these different corners fit into a single scheme. But the actual single scheme we do not know yet. So, the basic theoretical situation in LQG is simple and clear, not so in strings. Finally, no the predictions in LQG have not been changing. In fact, I wish there were solid predictions. There is none for the moment, like for strings. What exists in both cases is suggestion of possibilities. For instance strings suggested that perhaps the gravitational force could change at a measurable distance because of the extra dimensions, that supersymmetric particles be seen at lower energy, that black holes be formed at CERN. Nothing of this has been true so far, but this does not invalidate strings, because the theory is perfectly compatible with these effects existing but also compatible with these effects not existing. In very much the same way, it had been suggested that LQG might be combatible with violations of Lorentz invariance, and these seem to have been quite ruled out now by observations. But LQG is perfectly compatible with these Lorentz violation not being there. In fact, personally I have always been very skeptical at he suggestions that LQG might lead to Lorentz violation. If you read my old papers, I have always insisted that the theory is perfectly compatible with Lorentz invariance, and I could not see a source of Lorentz violation. These papers, for instance arXiv:gr-qc/0205108, which we wrote almost 10 years ago, indicate that LQG is locally Lorentz invariant, and were written long before the recent indications against Lorentz violation.
Regarding the compatibility between Loops and Strings, I really do not know. It is true that all the incompatibilities that Columbia indicates are there. But on the other hand we do not yet know the fundamental degrees of freedom of string theory. If there is a fundamental description of strings, this should be background independent; perhaps could it resemble somehow LQG? And in LQG there is no hint at the solution of the unification problem. Would it lead towards something more resembling strings? If I had to bet, I would say no, the two paths are really different, but I would not rule out the possibility a priori. I agree that the question is a bit premature, but I would say that the reason is not hat LQG is in a raw stage. LQG is a well and clearly defined theory, what is hard is to compute out of it. The reason is more because strings are in a raw stage, since we do not know a fundamental formulation of the theory, its basic degrees of freedom.
carlo rovelli