Toward Quantum Combinatorial Games presents the definition of a "quantum game", allowing a superposition of moves rather than a single classical move. This leaves me wondering: Since surreal numbers are a certain class of games, is there a similar class of "quantum surreal numbers" formed from quantum games, but richer than the surreals?

Toward Quantum Combinatorial Games presents the definition of a "quantum game", allowing a superposition of moves rather than a single classical move. This leaves me wondering: Since surreal numbers are a certain class of games, is there a similar class of "quantum surreal numbers" formed from quantum games, but richer than the surreals? What properties or structure do they have?