  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 $3$ markings - for all Hilbert schemes and for arbitrary curve
classes. In particular, for fixed $n$, the reduced quantum cohomologies of all
hyperk\"ahler varieties of $K3^{[n]}$-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 two CHL
Calabi-Yau threefold are Jacobi forms.
