  Let U be a universe on n elements, let k be a positive integer, and let F be
a family of (implicitly defined) subsets of U. We consider the problems of
partitioning U into k sets from F, covering U with k sets from F, and packing k
non-intersecting sets from F into U. Classically, these problems can be solved
via inclusion-exclusion in O*(2^n) time [BjorklundHK09]. Quantumly, there are
faster algorithms for graph coloring with running time O(1.9140^n) [ShimizuM22]
and for Set Cover with a small number of sets with running time O(1.7274^n
|F|^O(1)) [AmbainisBIKPV19]. In this paper, we give a quantum speedup for Set
Partition, Set Cover, and Set Packing whenever there is a classical enumeration
algorithm that lends itself to a quadratic quantum speedup, which, for any
subinstance on a subset X of U, enumerates at least one member of a
k-partition, k-cover, or k-packing (if one exists) restricted to (or projected
onto, in the case of k-cover) the set X in O*(c^{|X|}) time with c<2.
  Our bounded-error quantum algorithm runs in O*((2+c)^(n/2)) for Set
Partition, Set Cover, and Set Packing. When c<=1.147899, our algorithm is
slightly faster than O*((2+c)^(n/2)); when c approaches 1, it matches the
running time of [AmbainisBIKPV19] for Set Cover when |F| is subexponential in
n.
  For Graph Coloring, we further improve the running time to O(1.7956^n) by
leveraging faster algorithms for coloring with a small number of colors to
better balance our divide-and-conquer steps. For Domatic Number, we obtain a
O((2-\epsilon)^n) running time for some \epsilon>0.
