  For smooth complete intersections in the projective spaces, we use the
deformation invariance of Gromov-Witten invariants and results in classical
invariant theory to study the symmetric reduction of the WDVV equation by the
monodromy groups.
  For genus 0 invariants of non-exceptional Fano complete intersections other
than the cubic hypersurfaces, we find a square root recursion phenomenon. Based
on this we develop an algorithm to compute the genus 0 invariants of any
lengths and with any number of primitive insertions. The effectivity of our
algorithm in even dimension, and its validity in odd dimensions for invariants
with more than $\frac{\mathrm{rank}\ H^n(X)}{2}+2$ primitive insertions, is the
main conjecture of this paper.
  We prove the conjecture at the first non-trivial order, which amounts to
computation of genus zero Gromov-Witten invariants with exactly 4 primitive
insertions. For cubic hypersurfaces we complete the reconstruction by a
geometric approach. In particular, we obtain a reconstruction theorem for (i)
genus zero Gromov-Witten invariants of cubic hypersurfaces and odd dimensional
complete intersection of two quadrics and (ii) genus zero Gromov-Witten
invariants with at most 4 primitive insertions for all non-exceptional Fano
complete intersections of dimension at least 3. By the way, we obtain a
complete description of the cohomology ring of the Fano variety of lines on
smooth cubic hypersurfaces.
