  We identify emergence with the existence of states of potentiality related to
relevant physical quantities. We introduce the concept of 'potentiality state'
operationally and show how it reduces to 'superposition state' when standard
quantum mechanics can be applied. We consider several examples to illustrate
our approach, and define the potentiality states giving rise to emergence in
each example. We prove that Bell inequalities are violated by the potentiality
states in the examples, which, taking into account Pitowsky's theorem,
experimentally indicates the presence of quantum structure in emergence. In the
first example emergence arises because of the many ways water can be subdivided
into different vessels. In the second example, we put forward a full quantum
description of the Liar paradox situation, and identify the potentiality
states, which in this case turn out to be superposition states. In the example
of the soccer team, we show the difference between classical emergence as
stable dynamical pattern and emergence defined by a potentiality state, and
show how Bell inequalities can be violated in the case of highly contextual
experiments.
