Existence of CTCs implies also equivalence of quantum and classical computation (both in PSPACE).
Examples of games that are PSPACE-complete (when generalized so that they can be played on an n × n board) are the games hex and Reversi and the solitaire games Rush Hour, Mahjong, Atomix, and Sokoban.
Playing Mahjong solitaire optimally in the sense to maximize the probability of removing all tiles is PSPACE-complete, and the game gets NP-complete when peeking below tiles is allowed.