Anyway, so if nothing can produce anything more complex than itself, that implies that all the complexity of the universe was contained in the initial conditions of the Big Bang (actually it doesn't imply that, as we will see). Or at the very least that you could probably isolate some big cloud of early universe and say that it contains all the complexity of human society and everything that they ever produced. After all, this is what produced it, all you have to do is simulate it carefully enough and it will follow the same path.
Well, no, not really. I did gloss over something slightly enormous earlier, in modelling minds as Turing Machines. At least all the conscious human minds we know of are running on hardware (brains) that exist in this universe and therefore are subject to quantum uncertainty. Now, quantum uncertainty may not affect much on a macroscopic scale in an obvious way—but don't be too sure about that. And anyway, brains operate on a much smaller scale. Yeah, even our synapses are pretty big in quantum-mechanical terms, but not so much when you consider whether and how a molecule of neurotransmitter binds to its receptor.
I'm not going to go into the ineffable funkiness that is quantum mechanics; I would never do it justice. But we can say that at a macroscopic level, quantum effects are not less complicated than a random-number generator. Once it comes time to make a measurement, all you can say about a quantum state is that it has x% probability of turning out one way and y% probability of turning out another and z% probability for a third and so on. Once again, this doesn't do justice to what happens with quantum states; when they interact with each other it isn't probabilities they're altering, but “amplitudes,” which are complex numbers and thus can cancel out in ways regular randomness could never hope for. Still, if we treat every quantum observation as some sort of (possibly weighted) coin-flip then we certainly won't wind up with anything less complicated than the real thing.
In that case, all I've gone and done is changed the mind-model from Turing Machine to Non-Deterministic Turing Machine, and everyone knows (well, some people anyway) that a deterministic TM can do anything a non-deterministic TM can do, though perhaps not as fast (and even the added speed is not proven). Yeah, but I'm not talking ability, I'm talking complexity. Even though non-determinism doesn't add any real power in the sense of being able to do something formerly impossible, is a non-deterministic TM more complex than the deterministic one? Yes, it is, or can be. A non-deterministic TM can be modelled as a regular TM with an “oracle,” some source of random bits that determines which way the machine jumps at any non-deterministic choice, or alternatively an extra tape of input with the random bits on it. Wait, input? Input counts toward complexity! So a non-deterministic TM adds complexity on top of its deterministic counterpart, equal to the number of non-deterministic choices it had to make, or the amount of information it used from the oracle (approximately). In that case, quantum mechanics can add a huge amount of potential extra information and complexity. Just running the universe again from the exact same initial conditions wouldn't give you the same result, because quantum collapses would not necessarily happen the same way.
Now, I doubt that it has ever been shown that quantum randomness has a visible effect on our neurons, but I would have to admit it is possible. And generally, a lot of things we say are random, how many of them depend on quantum randomness if you look far enough? What is really still random if we discount quantum randomness? Brownian motion? Well, if you knew the precise positions and velocities of all the molecules in the area, and they all acted like classical Newtonian billiard balls, you could just plot their trajectories and collisions and figure it all out deterministically. Throwing a die? Well, if you knew the locations and velocities of everything precisely enough, you could model each bump and roll, each puff of minuscule air current and tiny angle, and find just where it gets tipped irrevocably into its final state. But you can't know the locations and velocities of every particle precisely enough. That's exactly quantum uncertainty. Brownian motion has to owe its randomness to quantum effects. I don't know if the same can be said of throwing a die; maybe that's just really hard Newtonian mechanics. But I wouldn't be surprised if it wound up being a quantum choice somewhere deep down far enough. I remember hearing once that even billiard balls don't act like billiard balls: a seven-ball combination is said to be impossible no matter how precise the equipment, due to quantum effects. There's so much going on there that quantum effects would be magnified enough. I don't know if that's true; I have to source for it.
People have speculated before that perhaps what gives consciousness its uniqueness, what makes it different from mechanical thinking (if indeed it is) and so hard to emulate in a computer, is quantum effects in our neurons. I certainly would not claim to know if that's true (maybe our minds really are just very complicated deterministic Turing Machines), but it seems to me that it's an idea that's worth considering.