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Abstract Given a set$$S=\{x^2+c_1,\dots,x^2+c_s\}$$defined over a field and an infinite sequence$$\gamma$$of elements ofS, one can associate an arboreal representation to$$\gamma$$, generalising the case of iterating a single polynomial. We study the probability that a random sequence$$\gamma$$produces a “large-image” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most setsSdefined over$$\mathbb{Z}[t]$$, and we conjecture a similar positive-probability result for suitable sets over$$\mathbb{Q}$$. As an application of large-image representations, we prove a density-zero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finite-index, and we classify allSpossessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.