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Polymeron comments on Thoughts on the Singularity Institute (SI) - Less Wrong
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Actually, with proper design, that can be made very quick and easy. You don't need to store the positions; you just need to store the states (win:black, win:white, draw - two bits per state).
The trick is, you store each win/loss state in a memory address equal to the 34-byte (or however long) binary number that describes the position in question. Checking a given state is then simply a memory retrieval from a known address.
I suspect that with memory on the order of 10^70 bytes, that might involve additional complications; but you're correct, normally this cancels out the complexity problem.