User blog comment:Wythagoras/My Turing machines/@comment-6768393-20131012123348/@comment-1605058-20131013145136

I really did not think about using notations to exactly express this number. But even then, I think such number would be too "chaotic" to be expressed in compact way (it will have high Kolgomorov complexity).

Strength of quantum computation relies, as far as I know, on parallelization, which is hard to do with direct simulation of machine. Tricks Ikosarakt mentioned are looking for patterns - we can parallelizise this process by looking for many patterns at the same time. But still, number of possible patterns certainly grows at superexponential rate, so quantum computing won't help much.

As we talk about diagonalizators, we have still more basic (zeroth?) level of diagonalizators, namely \(\Omega_1,\Omega_2,\Omega_3,...\). Also, where are Mahlo cardinals in this description?