The scientific community is becoming more and more interested in the research that applies the mathematical formalism of quantum theory to model human decision-making. In this paper, we expose the theoretical foundations of the quantum approach to cognition that we developed in Brussels. These foundations rest on the results of two decade studies on the axiomatic and operational-realistic approaches to the foundations of quantum physics. The deep analogies between the foundations of physics and those of cognition lead us to investigate the validity of quantum theory as a general and unitary framework for cognitive phenomena, and the empirical success of the Hilbert space models derived by such investigation provides a strong theoretical confirmation of this validity. However, two situations in the cognitive realm, “question order effects” and “response replicability,” indicate that even the Hilbert space framework could be insufficient to reproduce the expected pattern. This does not mean that the mentioned operational-realistic approach would be incorrect, but simply that a larger class of measurements would be in force in human cognition, so that an extended quantum formalism may be needed to deal with all of them. As we will explain, the recently derived “extended Bloch representation” of quantum theory (and the associated “general tension-reduction” model) precisely provides such extended formalism, while remaining within the same unitary interpretative framework.