Not quantum, but 5d chess with multiverse time travel exists. It sounds complicated (and it is) but it's actually pretty intuitive to actually pick up. Basically each piece moves as it traditionally does, but instead of two axes there are 4. So a rook can move in a straight line into the past, staying in the same position. That causes a time line branch with a new board with the time traveling rook. Bishops can move diagonally across time and multiverses.

I'm doing a terrible job explaining but this website has an excellent tutorial.

https://5d-chess.fandom.com/wiki/Tutorial