Interesting SUBSET-SUM problems
The main credit should go to John Fearnley!

Here is a PSPACE-complete problem given in (John Fearnley, Marcin Jurdzinski: Reachability in Two-Clock Timed Automata Is PSPACE-Complete. ICALP (2) 2013: 212-223):

\begin{equation}
 \mathtt{SUBSETSUM\mbox{-}GAME}=\{ S~ \forall(a_1 , b_1) \exists(e_1,f_1) \cdots \forall(a_n , b_n) \exists(e_n,f_n)  \},
\end{equation}
where 


$S$ and each $ a_i,b_i,e_i $, and $f_i$ are natural numbers ($1 \leq i \leq n$); and,
for every $ x=(x_1,\ldots,x_n) \in \times_{i=1}^n  \{ a_i,b_i \} $, there exists a $ y=(y_1,\ldots,y_n) \in \times_{i=1}^n  \{ e_i,f_i \} $ such that $ S = \sum_{i=1}^n = x_i+y_i $.


Similarly, we can define some complete problems for each level of polynomial hierarchy (PH). But, of course, in case of being complete at some level of PH, we need to release the condition of having only two natural numbers after each quantifier.