  The notion of 2--monoidal category used here was introduced by B.~Vallette in
2007 for applications in the operadic context. The starting point for this
article was a remark by Yu. Manin that in the category of quadratic algebras
(that is, "quantum linear spaces") one can also define 2--monoidal structure(s)
with rather unusual properties. Here we give a detailed exposition of these
constructions, together with their generalisations to the case of quadratic
operads.
  Their parallel exposition was motivated by the following remark. Several
important operads/cooperads such as genus zero quantum cohomology operad, the
operad classifying Gerstenhaber algebras, and more generally, (co)operads of
homology/cohomology of some topological operads, start with collections of
quadratic algebras/coalgebras rather than simply linear spaces.
  Suggested here enrichments of the categories to which components of these
operads belong, as well of the operadic structures themselves, might lead to
the better understanding of these fundamental objects.
  New version includes minor editorial changes and new references.
