  We construct spaces of quantum increasing sequences, which give quantum
families of maps in the sense of Soltan. We then introduce a notion of quantum
spreadability for a sequence of noncommutative random variables, by requiring
their joint distribution to be invariant under taking quantum subsequences. Our
main result is a free analogue of a theorem of Ryll-Nardzewski: for an infinite
sequence of noncommutative random variables, quantum spreadability is
equivalent to free independence and identical distribution with respect to a
conditional expectation.
