  The main purpose of this article is to study pro-$p$ groups with quadratic
$\mathbb{F}_p$-cohomology algebra, i.e. $H^\bullet$-quadratic pro-$p$ groups.
Prime examples of such groups are the maximal Galois pro-$p$ groups of fields
containing a primitive root of unity of order $p$.
  We show that the amalgamated free product and HNN-extension of
$H^\bullet$-quadratic pro-$p$ groups is $H^\bullet$-quadratic, under certain
necessary conditions. Moreover, we introduce and investigate a new family of
pro-$p$ groups that yields many new examples of $H^\bullet$-quadratic groups:
$p$-RAAGs. These examples generalise right angled Artin groups in the category
of pro-$p$ groups. Finally, we explore "Tits alternative behaviour" of
$H^\bullet$-quadratic pro-$p$ groups.
