Time is not the only resource. Space complexity counts the tape cells a machine uses — and because memory can be reused, space behaves very differently from time.
Definition · PSPACE
PSPACE is the class of languages decidable by a Turing machine using a polynomial amount of tape, with no limit on time.
P ⊆ NP ⊆ PSPACE ⊆ EXPTIMEMany two-player games are PSPACE-complete: deciding whether the first player has a winning strategy in generalized geography or Go-style games captures the alternation of “there exists a move such that for all replies…”.
Quantifiers as memory
TQBF — true quantified Boolean formulas — is the canonical PSPACE-complete problem, the space-bounded analogue of SAT.