  Most correlation inequalities for high-dimensional functions in the
literature, such as the Fortuin-Kasteleyn-Ginibre (FKG) inequality and the
celebrated Gaussian Correlation Inequality of Royen, are qualitative statements
which establish that any two functions of a certain type have non-negative
correlation. In this work we give a general approach that can be used to
bootstrap many qualitative correlation inequalities for functions over product
spaces into quantitative statements. The approach combines a new extremal
result about power series, proved using complex analysis, with harmonic
analysis of functions over product spaces. We instantiate this general approach
in several different concrete settings to obtain a range of new and
near-optimal quantitative correlation inequalities, including:
  $\bullet$ A quantitative version of Royen's celebrated Gaussian Correlation
Inequality. Royen (2014) confirmed a conjecture, open for 40 years, stating
that any two symmetric, convex sets must be non-negatively correlated under any
centered Gaussian distribution. We give a lower bound on the correlation in
terms of the vector of degree-2 Hermite coefficients of the two convex sets,
analogous to the correlation bound for monotone Boolean functions over
$\{0,1\}^n$ obtained by Talagrand (1996).
  $\bullet$ A quantitative version of the well-known FKG inequality for
monotone functions over any finite product probability space, generalizing the
quantitative correlation bound for monotone Boolean functions over $\{0,1\}^n$
obtained by Talagrand (1996). The only prior generalization of which we are
aware is due to Keller (2008, 2009, 2012), which extended Talagrand's result to
product distributions over $\{0,1\}^n$. We also give two different quantitative
versions of the FKG inequality for monotone functions over the continuous
domain $[0,1]^n$, answering a question of Keller (2009).
