  In this paper we study complexity of solving quadratic equations in the
lamplighter group. We give a complete classification of cases (depending on
genus and other characteristics of a given equation) when the problem is
$\mathbf{NP}$-complete or polynomial-time decidable. We notice that the
conjugacy problem can be solved in linear time. Finally, we prove that the
problem belongs to the class $\mathbf{XP}$.
