What is the most compact representation for a complete chess game state? Can it be as low as 64 bits?

I don't agree with the reasoning in either of the other two answers you've been given, but Shannon (1950) gave the number of chess positions as somewhere around 10^43, so 150+ bits. The position accounts for almost all of the state.

You could do better with variable-length encodings and perhaps represent all "interesting" "plausible" positions in 64 bits, but as others said, compactness is really not a top priority in representations for chess engines, it's much more important to be able to query and update the representation.

Sesse__

Shannon's estimate was based on very primitive methods; by generating random positions and using fairly advanced methods to see whether they are legal or not (ie., can you construct a proof game for it, or prove that it could never happen), you will get much closer. A group of people have been working on this, and their current best estimate is (4.822 +- 0.028) * 10^44, or a bit over 148 bits. (Amazingly enough, Shannon wasn't all that far off on this account! His estimated number of legal games seems much more dodgy, though.)

http://talkchess.com/forum3/viewtopic.php?f=7&t=77685&sid=e3...

Practically speaking, https://github.com/tromp/ChessPositionRanking gives a number between 0 and approx. 8.7 * 10^45 for any legal position, so it's only a couple of bits away from optimality.