Recently, two new spin-foam models have appeared in the literature, both
motivated by a desire to modify the Barrett-Crane model in such a way that the
imposition of certain second class constraints, called cross-simplicity
constraints, are weakened. We refer to these two models as the FKLS model, and
the flipped model. Both of these models are based on a reformulation of the
cross-simplicity constraints. This paper has two main parts. First, we clarify
the structure of the reformulated cross-simplicity constraints and the nature
of their quantum imposition in the new models. In particular we show that in
the FKLS model, quantum cross-simplicity implies no restriction on states. The
deeper reason for this is that, with the symplectic structure relevant for
FKLS, the reformulated cross-simplicity constraints, in a certain relevant
sense, are now \emph{first class}, and this causes the coherent state method of
imposing the constraints, key in the FKLS model, to fail to give any
restriction on states. Nevertheless, the cross-simplicity can still be seen as
implemented via suppression of intertwiner degrees of freedom in the dynamical
propagation. In the second part of the paper, we investigate area spectra in
the models. The results of these two investigations will highlight how, in the
flipped model, the Hilbert space of states, as well as the spectra of area
operators exactly match those of loop quantum gravity, whereas in the FKLS (and
Barrett-Crane) models, the boundary Hilbert spaces and area spectra are
different.Comment: 21 pages; statements about gamma limits made more precise, and minor
phrasing change