Does coNP-complete problem have subexponential size certificate? Assuming NP != coNP, then there is no polynomial size certificate for coNP-complete problem. But what about subexponential size certificate? Particularly for coSAT, is there subexponential size proof to prove a formula is unsatisfiable? If not, what is the negative evidence? Thanks
This is the topic of proof complexity, i.e. the size of certificates for the $\mathbf{co\text{-}NP}\text{-}complete$ problem $TAUT$ ($= coSAT$).

The short answer is: it is open.

On the negative side, we cannot even show that there are not polysize $Frege$ refutations for unsatisfiable formulas (let alone the general question of showing this for an arbitrary proof system, a propositional proof system can be thought as a nondeterministic algorithm for $TAUT$).

The question is also equivalent to $\mathbf{coNP} \subseteq NTime(2^{o(n)})$.