  We develop a new description of the much-studied $\kappa$-Minkowski
noncommutative spacetime, centered on representing on a single Hilbert space
not only the $\kappa$-Minkowski coordinates, but also the associated
differential calculus and the $\kappa$-Poincar\'e symmetry generators.
  In this "pregeometric" representation the relevant operators act on the
kinematical Hilbert space of the covariant formulation of quantum mechanics,
which we argue is the natural framework for studying the implications of the
step from commuting spacetime coordinates to the $\kappa$-Minkowski case, where
the spatial coordinates do not commute with the time coordinate. The
empowerment provided by this kinematical-Hilbert space representation allows us
to give a crisp characterization of the "fuzziness" of $\kappa$-Minkowski
spacetime, whose most striking aspect is a relativity of spacetime locality. We
show that relative locality, which had been previously formulated exclusively
in classical-spacetime setups, for a quantum spacetime takes the shape of a
dependence of the fuzziness of a spacetime point on the distance at which an
observer infers properties of the event that marks the point.
