  The formation of dark matter halos tends to occur anisotropically along the
filaments of the Cosmic Web, which induces both ellipticity-ellipticity (EE)
correlations between the shapes of halos, as well as ellipticity-direction (ED)
cross-correlations between halo shapes and the directions to neighboring halos.
We analyze the halo catalogue and the semi-analytic galaxy catalogue of the
recent Millennium Run Simulation to measure the EE and ED correlations
numerically at four different redshifts (z=0, 0.5, 1 and 2). For the EE
correlations, we find that (i) the major-axis correlation is strongest while
the intermediate-axis correlation is weakest; (ii) the signal is significant at
distances out to 10 Mpc/h; (iii) the signal decreases as z decreases; (iv) and
its behavior depends strongly on the halo mass scale, with larger masses
showing stronger correlations at large distances. For the ED correlations, we
find that (i) the correlations are much stronger than the EE correlations, and
are significant even out to distances of 50 Mpc/h; (ii) the signal also
decreases as z decreases; (iii) and it increases with halo mass at all
distances. We also provide empirical fitting functions for the EE and ED
correlations. The EE correlations are found to scale linearly with the linear
density correlation function, xi(r). While the ED cross-correlation is found to
scale as xi^{1/2}(r) at large distances beyond 10 Mpc/h. The best-fit values of
the fitting parameters for the EE and the ED correlations are all determined
through chi^{2}-statistics. Our results may be useful for quantifying the
filamentary distribution of dark matter halos over a wide range of scales.
