Squares in arithmetic progression over quadratic extensions of number fields

We study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus $5$ curve. Specifically, we determine the set of $K$-quadratic points on this curve under certain conditions on the base field $K$. Our main results rely on the algebraic properties of specific elliptic curves after performing a base change to suitable number fields. As a consequence, we establish that, under appropriate assumptions, any non-elementary arithmetic progression of five or six squares properly defined over a quadratic extension of $K$ must be of a specific form. Moreover, we prove the non-existence of such progressions of length greater than six under these assumptions.