  We study quantum causal structures in $1+1$ $\kappa$-Minkowski space-time
described by a Lorentzian Spectral Triple whose Dirac operator is built from a
natural set of twisted derivations of the $\kappa$-Poincar\'e algebra. We show
that the Lorentzian Spectral Triple must be twisted to accommodate the twisted
nature of the derivations. We exhibit various interesting classes of causal
functions, including an analog of the light-cone coordinates. We show in
particular that the existence of a causal propagation between two pure states,
the quantum analogs of points, can exist provided quantum constraints, linking
the momentum and the space coordinate, are satisfied. One of these constraints
is a quantum analog of the speed of light limit.
