  We introduce difference operators on the space of symmetric functions which
are a natural generalization of the $(q,t)$-Macdonald operators. In the
$t\to\infty$ limit, they satisfy the $A_{N-1}$ quantum $Q$-system. We identify
the elements in the spherical $A_{N-1}$ DAHA which are represented by these
operators, as well as within the quantum toroidal algebra of $gl_1$ and the
elliptic Hall algebra. We present a plethystic, or bosonic, formulation of the
generating functions for the generalized Macdonald operators, which we relate
to recent work of Bergeron et al. Finally we derive constant term identities
for the current that allow to interpret them in terms of shuffle products. In
particular we obtain in the $t\to\infty$ limit a shuffle presentation of the
quantum $Q$-system relations.
