  In network communication, it is common in broadcasting scenarios for there to
exist a hierarchy among receivers based on information they decode due, for
example, to different physical conditions or premium subscriptions. This
hierarchy may result in varied information quality, such as higher-quality
video for certain receivers. This is modeled mathematically as a degraded
message set, indicating a hierarchy between messages to be decoded by different
receivers, where the default quality corresponds to a common message intended
for all receivers, a higher quality is represented by a message for a smaller
subset of receivers, and so forth. We extend these considerations to quantum
communication, exploring three-receiver quantum broadcast channels with two-
and three-degraded message sets. Our technical tool involves employing quantum
non-unique decoding, a technique we develop by utilizing the simultaneous
pinching method. We construct one-shot codes for various scenarios and find
achievable rate regions relying on various quantum R\'enyi mutual information
error exponents. Our investigation includes a comprehensive study of pinching
across tensor product spaces, presenting our findings as the asymptotic
counterpart to our one-shot codes. By employing the non-unique decoding, we
also establish a simpler proof to Marton's inner bound for two-receiver quantum
broadcast channels without the need for more involved techniques. Additionally,
we derive no-go results and demonstrate their tightness in special cases.
