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Given a set 
$S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence 
$\gamma$ of elements of S, 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 sets S defined 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 all S possessing a particular kind of obstruction that generalises the post-critically finite case in single-polynomial iteration.
