  Noise correlations, such as those observable in the time of flight images of
a released cloud, are calculated for hard-core bosonic (HCB) atoms. We find
that the standard mapping of HCB systems onto spin-1/2 XY models fails in
application to computation of noise correlations. This is due to the
contribution of multiply occupied virtual states to noise correlations in
bosonic systems. Such states do not exist in spin models. We use these
correlations to explore quantum coherence of the ground states and re-address
the relationship between the peaks present in noise correlation and the Mott
phase. Our analysis points to distinctive new experimental signatures of the
Mott phase. The importance of these correlations is illustrated in an example
of a quasiperiodic potential that exhibits a localization transition. In this
case, in contrast to the momentum distribution, the noise correlations reveal
the presence of quasiperiodic order in the localized phase.
