Uncountability / Diagonalization

In this game, the coins were flipped. The outcome can be thought of as an infinite sequence over {H, T} that eventually alternates (with H at odd indices and T at even indices, as below).

Are there countably infinitely many or uncountably infinitely many such sequences for the game?

Explain/prove your answer.

Some sample games of Quaboom:

HTHTHTHTHTHTHTHT. . . HTHTHTHTHTHTHTHT. . .

THTHTHTHTHTHTHTH. . . HTHTHTHTHTHTHTHT. . .

HHHHHHHHHHHHHHHH. . . HTHTHTHTHTHTHTHT. . .

TTTTTTTTTTTTTTTT. . . HTHTHTHTHTHTHTHT. . .

TTTHTHTTTHTHTHTT. . . HTHTHTHTHTHTHTHT. . .

HHHTTHTHTHTHHTHT. . . HTHTHTHTHTHTHTHT. . .