The Wick-Rotation rotates imaginary time into inverse temperature (as can be seen from its "rotating" the Schrödinger equation into the heat equation). Now since entropy is temperature's conjugate, I was wondering what its Wick-Rotation relates to?
Basically, he defines what he calls the "quantropy", which is just the classical entropy formula with $\beta$ replaced by $-i/\hbar$ and the energy replaced with the action. (Note that this is not at all the same as the von Neumann entropy.) The quantropy is essentially the Wick rotation of the entropy.
He then shows that finding a stationary point of the quantropy gives you the Schrödinger equation (and various other aspects of quantum mechanics), in much the same way that maximising the entropy gives you the Boltzmann distribution and much of the rest of classical statistical mechanics. It's quite interesting.