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This paper studies quality of service (QoS) routing for quorumcast, a generalized form of multicast communication. The need of quorumcast communication arises in a number of distributed applications such as distributed synchronization and resource discovery, but little work has been done on routing quorumcast messages with or without QoS constraints. Moreover, most known research work aimed to construct a minimum cost tree that spans the source and the quorumcast group members. In this paper, we further consider the path quality of a tree in terms of source-destination delay constraints imposed by applications that use the tree. As the delay-constrained quorumcast routing problem is NP-complete, we propose an efficient heuristic QoS routing algorithm. We also consider how to detect and remove loops in the course of tree construction and how to deal with member join/leave in/from the quorumcast pool. Our simulation study shows that the proposed algorithm performs well and constructs a quorumcast tree with the cost close to that of the "optimal" routing tree.