Consequences of a distillation algorithm for PSPACE The following notion of a distillation algorithm comes from "On Problems Without Polynomial Kernels".


  Let a language $L$ be given.  A distillation algorithm for $L$ takes a given
  list of input strings $\{ x_i \}_{i \in [t]}$ and computes an output string
  $y$ such that:
  
  (1) $y \in L$ if and only if there exists $i \in [t]$ such that $x_i \in L$
  
  (2) $\vert y \vert \leq p(Max_{i\in[t]} \vert x_i \vert)$ for some polynomial $p$
  
  (3) The algorithm computes $y$ in at most $q(\sum_{i\in[t]}\vert x_i \vert)$ time for some polynomial $q$


It has been shown that if there exists a distillation algorithm for an $NP$-complete problem, then $coNP \subseteq NP/poly$.  Moreover, $PH = \Sigma_3$.


  See details and discussion in:
  
  
  "Infeasibility of Instance Compression and Succinct PCPs for NP"
  "On Problems Without Polynomial Kernels"
  "Lower bounds on kernelization"
  


Questions:


Could there exist a distillation algorithm for a $PSPACE$-complete
problem?
If such an algorithm existed, what complexity consequences would we
get?
Theorem 15.3 of the recent "Parameterized Algorithms" textbook by Cygan et al. states the following:

"Let $L, R ⊆ \Sigma^*$ be two languages. If there exists an OR-distillation of L into R, then $L\in coNP / poly$"

So, I think that if there exists an OR-distillation from a PSPACE-complete language $L$ to itself, then $PSPACE \subseteq coNP/poly$, i.e. not only does the polynomial-hierarchy collapse, but also PSPACE collapses with it.