  Let $X$ be a curve of genus $g>1$ over $\mathbb{Q}$ whose Jacobian $J$ has
Mordell--Weil rank $r$ and N\'eron--Severi rank $\rho$. When $r < g+ \rho - 1$,
the geometric quadratic Chabauty method determines a finite set of $p$-adic
points containing the rational points of $X$. We describe algorithms for
geometric quadratic Chabauty that translate the geometric quadratic Chabauty
method into the language of $p$-adic heights and $p$-adic (Coleman) integrals.
This translation also allows us to give a comparison to the (original)
cohomological method for quadratic Chabauty. We show that the finite set of
$p$-adic points produced by the geometric method is contained in the finite set
produced by the cohomological method, and give a description of their
difference.
