  The Beilinson-Bloch type conjectures predict that the low degree rational
Chow groups of intersections of quadrics are one dimensional. This conjecture
was proved by Otwinowska. Making use of homological projective duality and the
recent theory of (Jacobians of) noncommutative Chow motives, we give an
alternative proof of this conjecture in the case of a complete intersection of
either two quadrics or three odd-dimensional quadrics. Moreover, without the
use of the powerful Lefschetz theorem, we prove that in these cases the unique
non-trivial algebraic Jacobian is the middle one. As an application, making use
of Vial's work, we describe the rational Chow motives of these complete
intersections and show that smooth fibrations in such complete intersections
over small dimensional bases S verify Murre's conjecture (dim(S) less or equal
to 1), Grothendieck's standard conjectures (dim(S) less of equal to 2), and
Hodge's conjecture (dim(S) less or equal to 3).
