  Contradictory results have been reported on the time evolution of the
alignment between clusters and their Brightest Cluster Galaxy (BCG). We study
this topic by analyzing cosmological hydro-simulations of 24 massive clusters
with $M_{200}|_{z=0} \gtrsim 10^{15}\, M_\odot$, plus 5 less massive with $1
\times 10^{14} \lesssim M_{200}|_{z=0} \lesssim 7 \times 10^{14}\, M_\odot$,
which have already proven to produce realistic BCG masses. We compute the BCG
alignment with both the distribution of cluster galaxies and the dark matter
(DM) halo. At redshift $z=0$, the major axes of the simulated BCGs and their
host cluster galaxy distributions are aligned on average within 20$^\circ$. The
BCG alignment with the DM halo is even tighter. The alignment persists up to
$z\lesssim2$ with no evident evolution. This result continues, although with a
weaker signal, when considering the projected alignment. The cluster alignment
with the surrounding distribution of matter ($3R_{200}$) is already in place at
$z\sim4$ with a typical angle of $35^\circ$, before the BCG-Cluster alignment
develops. The BCG turns out to be also aligned with the same matter
distribution, albeit always to a lesser extent. These results taken together
might imply that the BCG-Cluster alignment occurs in an outside-in fashion.
Depending on their frequency and geometry, mergers can promote, destroy or
weaken the alignments. Clusters that do not experience recent major mergers are
typically more relaxed and aligned with their BCG. In turn, accretions closer
to the cluster elongation axis tend to improve the alignment as opposed to
accretions closer to the cluster minor axis.
