  Let $E$ be an elliptic curve over the rational field $\mathbf{Q}$. Let $K$ be
a quadratic extension over $\mathbf{Q}$. Let $\mathrm{ST}(E/K)$ dente the
Shafarevich-Tate group of $E$ over $K$. We show that (under mild conditions on
$E$) for every $r>0$, there are infinitely many quadratic twists
$E^d/\mathbf{Q}$ of $E/\mathbf{Q}$ such that
$\mathrm{dim}_{\mathbf{F}_2}(\mathrm{ST}(E^d/K)[2]) > r$
