Lower Bounds from Succinct Hitting Sets

We investigate the consequences of the existence of ``efficiently describable'' hitting sets for polynomial sized algebraic circuit ($\mathsf{VP}$), in particular, \emph{$\mathsf{VP}$-succinct hitting sets}. Existence of such hitting sets is known to be equivalent to a ``natural-proofs-barrier'' towards algebraic circuit lower bounds, from the works that introduced this concept (Forbes \etal (2018), Grochow \etal (2017)). We show that the existence of $\mathsf{VP}$-succinct hitting sets for $\mathsf{VP}$ would either imply that $\mathsf{VP} \neq \mathsf{VNP}$, or yield a fairly strong lower bound against $\mathsf{TC}^0$ circuits, assuming the Generalized Riemann Hypothesis (GRH). This result is a consequence of showing that designing efficiently describable ($\mathsf{VP}$-explicit) hitting set generators for a class $\mathcal{C}$, is essentially the same as proving a separation between $\mathcal{C}$ and $\mathsf{VPSPACE}$: the algebraic analogue of \textsf{PSPACE}. More formally, we prove an upper bound on \emph{equations} for polynomial sized algebraic circuits ($\mathsf{VP}$), in terms of $\mathsf{VPSPACE}$. Using the same upper bound, we also show that even \emph{sub-polynomially explicit hitting sets} for $\mathsf{VP}$ -- much weaker than $\mathsf{VP}$-succinct hitting sets that are almost polylog-explicit -- would imply that either $\mathsf{VP} \neq \mathsf{VNP}$ or that $\mathsf{P} \neq \mathsf{PSPACE}$. This motivates us to define the concept of \emph{cryptographic hitting sets}, which we believe is interesting on its own.