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We study whether in-medium showers of high energy gluons can be treated as a series of independent splitting processes, or if there is significant quantum overlap between consecutive splittings. Accounting for the Landau-Pomeranchuk-Migdal (LPM) effect, we calculate such overlap effects to leading order in high energy \alpha_{s}(\mu), for the simplest theoretical situation of a static, homogeneous, infinite medium. We then investigate a measure of overlap effects that is independent of physics that can be absorbed into an effective value of the jet-quenching parameter q-hat.