TQBF PSPACE-COMPLETE : Why this algorithm is exponential but Savitch's not?
In the proof of the $\textit{PSPACE-}$completeness of TQBF, we need to construct a formula of some specific type. The recursive midpoint algorithm is easy to implement (and Savitch's Theorem requires finding the $\textit{algorithm}$), while the same idea, if performed in a similar manner, results in an exponential formula (and the completeness proof requires the $\textit{formula}$). 

More generally, suppose there's some language $\textit{L}$, an optimal algorithm $A$ deciding $L$, and the smallest formula (maybe quantified) $\varphi$, s.t. $\varphi(x) = 1 \Leftrightarrow x \in L$. For me, it's not very clear why $A\in \textit{PSPACE}$ would imply that  $\varphi$ is small (I only see it as a consequence of the $\textit{PSPACE}-$completeness theorem)