What's wrong with this proof that NP=Co-NP implies NP=PSPACE
A totally quantified boolean formula consists of an alternating sequence of quantifiers. This sequence is not of length 2, but arbitrarily long. That is the mistake you're making: an instance can be represented as a pair (P, phi) where phi is a propositional formula and L = (X_1, ..., X_k) is a partition of the set of variables occuring in phi, such that X_i are the variables in the i'th quantifier (which is an existential one if i is odd, and universal otherwise). You are assuming that this instance is equivalent to (L', phi), L' = (X_1', X_2') where X_1' is the union of the odd-indexed X_i, and X_2' is the union of the even-indexed X_i. Please verify for yourself that this is not necessarily true.