  The (3+1)-dimensional $\kappa$-(A)dS noncommutative spacetime is explicitly
constructed by quantizing its semiclassical counterpart, which is the
$\kappa$-(A)dS Poisson homogeneous space. This turns out to be the only
possible generalization of the well-known $\kappa$-Minkowski spacetime to the
case of non-vanishing cosmological constant, under the condition that the time
translation generator of the corresponding quantum (A)dS algebra is primitive.
Moreover, the $\kappa$-(A)dS noncommutative spacetime is shown to have a
quadratic subalgebra of local spatial coordinates whose first-order brackets in
terms of the cosmological constant parameter define a quantum sphere, while the
commutators between time and space coordinates preserve the same structure of
the $\kappa$-Minkowski spacetime. When expressed in ambient coordinates, the
quantum $\kappa$-(A)dS spacetime is shown to be defined as a noncommutative
pseudosphere.
