  In singularity theory or algebraic geometry, it is natural to investigate
possible Hilbert functions for special algebras $A$ such as local complete
intersections or more generally Gorenstein algebras. The sequences that occur
as {the} Hilbert functions of standard graded complete intersections are well
understood classically thanks to Macaulay and Stanley. Very little is known in
the local case except in codimension two. In this paper we characterise the
Hilbert functions of quadratic Artinian complete intersections of codimension
three. Interestingly we prove that a Hilbert function is admissible for such a
Gorenstein ring if and only if is admissible for such a complete intersection.
We provide an effective construction of a local complete intersection for a
given Hilbert function. We prove that the symmetric decomposition of such a
complete intersection ideal is determined by its Hilbert function.
