I think the idea is that you can't pick a different interpretation for the rock implementing a specific computation for each instant of time. A convincing narrative of the physical processes in a rock instantiating a consciousness would require a mapping from rock states to the computational process of the consciousness that remains stable over time. With the physical processes going on in rocks being pretty much random, you wouldn't get the moment-to-moment coherence you'd need for this even if you can come up with interpretations for single instants.

One intuition here is that once you come up with a good interpretation, the physical system needs to be able to come up with correct results from computations that go on longer than where you extrapolated doing your interpretation. If you try to get around the single instant thing and make a tortured interpretation of rock states representing the computation of, say, 100 consecutive computations of the consciousness, the interpretation is going to have the rock give you garbage for computation 101. You're just doing the computation yourself now and painstakingly fitting things to random physical noise in the rock.

Try bisimulation.

A homomorphism is a "structure preserving map", and is quite general until you specify what is preserved.

From my brief reading of Chalmers, he's basically captured my objection. As Risto_Saarelma says, the point is that a mapping merely of states should not count. As long as the sets of object states are not overlapping, there's a mapping into the abstract computation. That's boring. To truly instantiate the computation, what has to be put in is the causal structure, the rules of the computation, and these seem to be far more restrictive than one trace of possible states.

Chalmer's "clock and dial" seems to get around this in that it can enumerate all possible traces, which seems to be equivalent to capturing the rules, but still feels decidedly wrong.