Possible quantum states and eternity According to a TED talk by Sean Carroll (about 12' 40" in) if our universe is going to expand forever according the observed expansion rate we have calculated and the possible quantum states for our universe will occur in time span of $10^{10^{120}}$ years. The question is what would happen after that time span would pass? 
 A: This number looks to be the Poincare recurrence time. In a world with $10^{90}$ particles the Poincare recurrence time is about $T~\simeq~10^{10^{90}}$ time units. The time units are comparable to the Planck time. This is similar to the number quoted here. This is also comparable to the stability of the de Sitter vacuum. The question is then; What happens next? That is not certain, but it may be the cosmos quantum fluctuates into a new cosmology. 
There is also another recurrence time. This is the time it takes for the quantum states to return to the phase they had at some prior time. This is exponentially larger, $T~\simeq~10^{10^{10^{90}}}$ seconds. All of this may have some connection to multiverse, and the physics of duplication or near duplication of quantum states. If you were able to travel $10^{10^{90}}$ Planck units of distance out you might encounter a copy of the world here. These worlds though might be many world quantum alternative worlds to our world. The recurrence of the quantum phase might have something to do with the number of possible world branches in the yggdrasillian set of future possibilities.
Of course with this speculation there may never be any way to verify any of it. The many worlds interpretation of quantum mechanics is like all other interpretations likely not really testable. By the same there is no way we can observe anything that far in the distance, even if space is an infinite $\mathbb R^3$ space.
A: Sean Carroll's talk is based on the idea that dark energy is, or behaves like, a cosmological constant and therefore in the far future the universe will be well described by the de Sitter geometry. Exactly what this means is somewhat involved, but for Carroll's talk the significance is that in a de Sitter universe the size of the observable universe is constant. For an observer in a de Sitter universe the bit of the universe they can see is surrounded by a horizon that is similar to a black hole event horizon, and this horizon is at a fixed distance that doesn't change with time.
The second key idea, which Sean Carroll doesn't mention in the talk, is that there is an idea called the holographic principle that states the maximum number of possible configurations of a system is proportional to its surface area. Specifically, the number of configurations is equal to the area in Planck units. This idea arose from studies on black hole entropy, though we should emphasise that the idea is poorly understood and highly speculative.
Anyhow, if you accept both the ideas discussed above then the maximum possible number of states of a de Sitter universe is the area of its horizon. Based on what we know about our universe we expect its de Sitter horizon to have a radius of about $10^{60}$ Plank distances, so the area will be about $\pi$ times this radius squared or about $10^{120}$ Planck areas (this is such a vague calculation we can simply ignore the factor of $\pi$).
And finally we can see where Carroll got his number $10^{10^{120}}$. If we have an observable universe with a fixed size, so it has a fixed maximum possible number of states $10^{120}$, then the amount of time taken for the universe to explore all these configurations is about ten to the power of the number of states or:
$$ T = 10^{10^{120}} $$
But don't take all this seriously because it depends on all sorts of ideas and assumptions that are exceedingly speculative to say the least. The point of my answer is purely to explain how Carroll got that magic number - it doesn't mean I believe a word of it!
