  We exhibit families of $4$-CNF formulas over $n$ variables that have
sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) $d$
but require SOS proofs of size $n^{\Omega(d)}$ for values of $d = d(n)$ from
constant all the way up to $n^{\delta}$ for some universal constant$\delta$.
This shows that the $n^{O(d)}$ running time obtained by using the Lasserre
semidefinite programming relaxations to find degree-$d$ SOS proofs is optimal
up to constant factors in the exponent. We establish this result by combining
$\mathsf{NP}$-reductions expressible as low-degree SOS derivations with the
idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and
Riis'03], and then applying a restriction argument as in [Atserias, M\"uller,
and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a
generic method of amplifying SOS degree lower bounds to size lower bounds, and
also generalizes the approach in [ALN14] to obtain size lower bounds for the
proof systems resolution, polynomial calculus, and Sherali-Adams from lower
bounds on width, degree, and rank, respectively.
