Non-uniform lower bounds from uniform
MA $\subseteq$ PSPACE $\subseteq$ EXP $\subseteq$ NEXP ​ and ​ P/poly has undecidable unary languages, so

NEXP $\subseteq$ P/poly ​ $\implies$ ​ NEXP $\subset$ P/poly ​ $\implies$ ​ NEXP $\subseteq$ EXP .

As something like a generalization of that ​ (showing that EXP-complete languages are

in CHECK is difficult) , ​ for all languages L in CHECK with a checker whose queries all have

length at most ​ nd , ​ for all non-decreasing functions ​ S : {0,1,2,3,4,...} $\to$ {0,1,2,3,4,...} :

If L's circuit-size in bits is at most S then

L ​ $\in$ ​ DSPACE$\left(\hspace{-0.04 in}\left(\hspace{-0.02 in}n^d\hspace{-0.05 in}\cdot \hspace{-0.03 in}S\hspace{-0.05 in}\left(\hspace{-0.03 in}n^d\hspace{-0.02 in}\right)\hspace{-0.04 in}\right)\hspace{-0.04 in}+\hspace{-0.03 in}poly\hspace{.02 in}(n)\hspace{-0.03 in}\right)$ ​ $\subseteq$ ​ DTIME$\hspace{-0.02 in}\left(\hspace{-0.05 in}2^{\left(n^{\hspace{.02 in}d} \hspace{-0.02 in}\cdot S\left(\hspace{-0.02 in}n^{\hspace{.02 in}d}\hspace{-0.03 in}\right)\hspace{-0.03 in}\right)+\hspace{.02 in}poly\hspace{.02 in}(n)}\hspace{-0.04 in}\right)$ ​ ​ ,

so by contrapositive, if ​ ​ ​ L ​ $\not\in$ ​ DTIME$\hspace{-0.02 in}\left(\hspace{-0.05 in}2^{\left(n^{\hspace{.02 in}d} \hspace{-0.02 in}\cdot S\left(\hspace{-0.02 in}n^{\hspace{.02 in}d}\hspace{-0.03 in}\right)\hspace{-0.03 in}\right)+\hspace{.02 in}poly\hspace{.02 in}(n)}\hspace{-0.04 in}\right)$

then L's circuit-size in bits exceeds S infinitely-often.

("in bits" as as opposed to "in gates".)

Similarly, for all functions T and $\ell$ and $S$ from {0,1,2,3,...} to {1,2,3,4,5,...},

for all search problems R in ​ FNTIME(T) ​ whose output-lengths are bounded above by $\ell$:

If R's circuit-size in bits is at most S then R is solvable by ​ FDTIME$\left(\hspace{-0.02 in}2^{S(n)} \hspace{-0.04 in}\cdot \hspace{-0.04 in}poly\hspace{.02 in}(S(n))\hspace{-0.04 in}\cdot \hspace{-0.04 in}T(n)\hspace{-0.03 in}\right)$ ,

so by contrapositive, if R is not solvable by ​ FDTIME$\left(\hspace{-0.02 in}2^{S(n)} \hspace{-0.03 in}\cdot \hspace{-0.03 in}\left(\left(\hspace{.05 in}poly\hspace{.02 in}(S(n)) \cdot \ell(n)\right)+ T(n)\right)\hspace{-0.03 in}\right)$

then R's circuit-size in bits exceeds S infinitely-often.