  In this paper we consider the relationship between monomial-size and
bit-complexity in Sums-of-Squares (SOS) in Polynomial Calculus Resolution over
rationals (PCR/$\mathbb{Q}$). We show that there is a set of polynomial
constraints $Q_n$ over Boolean variables that has both SOS and PCR/$\mathbb{Q}$
refutations of degree 2 and thus with only polynomially many monomials, but for
which any SOS or PCR/$\mathbb{Q}$ refutation must have exponential
bit-complexity, when the rational coefficients are represented with their
reduced fractions written in binary.
