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 NPOn Problems Without Polynomial KernelsLower bounds on kernelizationQuestions:Could there exist a distillation algorithm for a $PSPACE$-completeproblem?If such an algorithm existed, what complexity consequences would weget?
Consequences of a distillation algorithm for PSPACE
cc.complexity theory;np hardness;parameterized complexity;polynomial hierarchy;pspace
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.