  The constraint satisfaction problem (CSP) and its quantified extensions,
whether without (QCSP) or with disjunction (QCSP_or), correspond naturally to
the model checking problem for three increasingly stronger fragments of
positive first-order logic. Their complexity is often studied when
parameterised by a fixed model, the so-called template.
  It is a natural question to ask when two templates are equivalent, or more
generally when one "contain" another, in the sense that a satisfied instance of
the first will be necessarily satisfied in the second. One can also ask for a
smallest possible equivalent template: this is known as the core for CSP.
  We recall and extend previous results on containment, equivalence and
"coreness" for QCSP_or before initiating a preliminary study of cores for QCSP
which we characterise for certain structures and which turns out to be more
elusive.
