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We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in theSum-of-Squares (SoS) proof system. Our main result is that for every Booleanfunction $f: \{0,1\}^n \rightarrow \{0,1\}$, SoS requires degree$\Omega(s^{1-\epsilon})$ to prove that $f$ does not have circuits of size $s$(for any $s > \mathrm{poly}(n)$). As a corollary we obtain that there are nolow degree SoS proofs of the statement NP $\not \subseteq $ P/poly.  We also show that for any $0 < \alpha < 1$ there are Boolean functions withcircuit complexity larger than $2^{n^{\alpha}}$ but SoS requires size$2^{2^{\Omega(n^{\alpha})}}$ to prove this. In addition we prove analogousresults on the minimum \emph{monotone} circuit size for monotone Boolean slicefunctions.  Our approach is quite general. Namely, we show that if a proof system $Q$ hasstrong enough constraint satisfaction problem lower bounds that only depend ongood expansion of the constraint-variable incidence graph and, furthermore, $Q$is expressive enough that variables can be substituted by local Booleanfunctions, then the MCSP problem is hard for $Q$.

Sum-of-Squares Lower Bounds for the Minimum Circuit Size Problem