Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface

We conjecture that the generating series of Gromov-Witten invariants of the Hilbert schemes of n points on a K3 surface are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture in genus 0 and for at most three markings - for all Hilbert schemes and for arbitrary curve classes. In particular, for fixed n, the reduced quantum cohomologies of all hyperk & auml;hler varieties of K3 OE n]-type are determined up to finitely many coefficients. As an application we show that the generating series of 2-point Gromov-Witten classes are vector-valued Jacobi forms of weight- 10, and that the fiberwise Donaldson-Thomas partition functions of an order-2 CHL Calabi-Yau threefold are Jacobi forms.