  The quantum homology of the monotone complex quadric surface splits into the
sum of two fields. We outline a proof of the following statement: The unities
of these fields give rise to distinct symplectic quasi-states defined by
asymptotic spectral invariants. In fact, these quasi-states turn out to be
"supported" on disjoint Lagrangian submanifolds. Our method involves a spectral
sequence which starts at homology of the loop space of the 2-sphere and whose
higher differentials are computed via symplectic field theory, in particular
with the help of the Bourgeois-Oancea exact sequence.
