  We present measurements of the triaxial dark matter halo shapes and alignment
correlation functions in the Millennium and Millennium-2 dark matter N-body
simulations. These two simulations allow us to measure the distributions of
halo shapes down to 10% of the virial radius over a halo mass range of 6E9 -
2E14 M_sun/h. We largely confirm previous results on the distributions of halo
axis ratios as a function of halo mass, but we find that the median angle
between halo major axes at different halo radii can vary by a factor of 2
between the Millennium-1 and 2 simulations because of the different mass
resolution. Thus, error in the shape determinations from limited resolution is
potentially degenerate with the misalignment of halo inner and outer shapes
used to constrain Brightest Cluster Galaxy alignments in previous works. We
also present simplifying parameterizations for the 3-D halo-mass alignment
correlation functions that are necessary ingredients for triaxial halo models
of large-scale structure and models of galaxy intrinsic alignments as
contaminants for cosmic shear surveys. We measure strong alignments between
halos of all masses and the surrounding dark matter overdensities out to
several tens of Mpc/h, in agreement with observed shear-galaxy and cluster
shape correlations. We use these measurements to forecast the contribution to
the weak lensing signal around galaxy clusters from correlated mass along the
line-of-sight. For prolate clusters with major axes aligned with the
line-of-sight the fraction of the weak lensing signal from mass external to the
cluster can be twice that predicted if the excess halo alignment correlation is
assumed to be zero.
