  The AdS boundary correlators and their dual correlation functions of boundary
operators have been the main dynamic observables of the holographic duality
relating a bulk AdS theory and a boundary conformal field theory. We show that
tree-level AdS boundary correlators for generic states can be expressed as
nonlocal differential operators of a certain structure acting on contact Witten
diagrams. We further write the boundary correlators in a form that is very
similar to flat space amplitudes, with Mandelstam variables replaced by certain
combinations of single-state conformal generators, prove that all tree-level
AdS boundary correlators have a differential representation, and detail the
conversion of such differential expressions to position space. We illustrate
the construction through the computation of the boundary correlators of scalars
coupled to gluons and gravitons; when converted to position space, they
reproduce known results. Color-kinematics duality and BCJ relations can be
defined in analogy with their flat space counterparts, and are respected by the
scalar correlators with a gluon exchange. We also discuss potential approaches
to the double copy and find that its direct generalization may require
nontrivial extensions.
