1. Information: It has to have a large repertoire of accessible states, i.e., the ability to store a large amount of information.
2. Integration: This information must be integrated into a unified whole, i.e., it must be impossible to decompose the system into nearly independent parts, because otherwise these parts would subjectively feel like two separate conscious entities.
1) is reasonable, although it doesn't say much about how much memory is required. 2) is clearly bogus - we know that biological brains don't work that way. They have at least a hundred or so subunits handling different functions. Some are very specific, such as face recognition. Tononi says that in his paper: "different parts of the cerebral cortex are specialized for different functions, yet a vast network of connections allows these parts to interact profusely." So he admits it's not one big component; it's a network of components with explicit, limited connectivity.
Where Tonini is coming from is the concept of "qualia".[2] This is a philosophical concept, roughly "The 'what it is like' character of mental states. The way it feels to have mental states such as pain, seeing red, smelling a rose, etc." Toroni hypothesizes that qualia can be (are?) represented as points in a space of 2^n dimensions. He writes "Each axis is labeled with the probability p for that state, going from 0 to 1, so that a repertoire (i.e.,a probability distribution on the possible states of the complex) corresponds to a point in Q (Fig. 5)." This is similar to the internal state of an artificial neural network.
The physics paper conjectures that if only we had some materials (?) called "computronium" and "perceptronium", we could get work done in qualia space, through a process where energy is removed from the system in a way vaguely similar to simulated annealing. There's a lot of math and discussion around the idea that some randomness has to be applied to make the thing go; not enough, and it stalls out; too much, and it's just noise. There's a suggestion that certain forms of noise will make it work better. All this sound vaguely similar to the way the random initialization of ANNs matters a lot, and recent work has made ANNs better by choosing the random initial conditions in certain ways. Neither paper mentions what's going on in the ANN world, yet they seem to be approaching some of the same concepts. I'm not an ANN expert; someone in that field may have a better idea of whether the concepts are related.
All this is from the "consciousness is special and magic" crowd. Both authors assume that is the case, and try to reason from there. That's closer to theology than physics.
[1] http://www.biolbull.org/content/215/3/216.full.pdf+html [2] https://en.wikipedia.org/wiki/Qualia