We formulate the Kurepa hypothesis (KH) and its generalizations in determinacy and prove KH fails there. More generally, we show that if $\kappa < \Theta$ and $cof(\kappa) > \omega$, then the set of branches through a  $\kappa$-tree must be well-orderable and has cardinality less than or equal to \kappa. We also show there are no maximal almost disjoint families at $\omega_1$ and its appropriate generalizations at regular cardinals $\kappa < \Theta$. This is joint work with William Chan and Steve Jackson.