  The formalism of quantum theory in Hilbert space has been applied with
success to the modeling and explanation of several cognitive phenomena, whereas
traditional cognitive approaches were problematical. However, this 'quantum
cognition paradigm' was recently challenged by its proven impossibility to
simultaneously model 'question order effects' and 'response replicability'. In
Part I of this paper we describe sequential dichotomic measurements within an
operational and realistic framework for human cognition elaborated by
ourselves, and represent them in a quantum-like 'extended Bloch representation'
where the Born rule of quantum probability does not necessarily hold. In Part
II we apply this mathematical framework to successfully model question order
effects, response replicability and unpacking effects, thus opening the way
toward quantum cognition beyond Hilbert space.
