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A curious difference is that in the quantum case an error-correcting encoding is used, and we care about the representation (i.e., the vector space); in the Rubik's cube case, there is no encoding (I wonder what one would be?), and we don't care about the representation.
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I didn't mention my motivation, but may as well - I've wondered for decades what happens if you make the units of currency a non-Abelian group (rather than Abelian, as usually). And was just playing around with adversarial games that can be played in a non-Abelian economy :-)
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To go back to the original, though, it's fun to think about rules for actual adversarial cubing competitions. No longer about dexterity, but actual combat, more like chess.
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In general, if you use a shared ledger where a transaction applies some group element g to my account (recipient), and g^{-1} to yours (sender), that's pretty interesting.
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If the underlying group is the positive reals, with operation multiplication, then this is a reasonable approximation to standard currencies & a standard economy. If the group is, say, the invertible 2 x 2 matrices (or, say, with det 1), it's a decidedly nonstandard economy.
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I believe Abelian groups would (probably) give pretty standard variations on existing economies (though maybe with interesting twists). Non-Abelian groups would, I expect, be radically different.
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In particular, the commutator would be fascinating, a way of getting something out of pure temporal ordering. This seems like it might be useful as a co-operation mechanism?
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