Sarcasm detection in conversation, a theoretically and practically challenging artificial intelligence task, aims to discover elusively ironic, contemptuous, and metaphoric information implied in daily conversations. Most of the recent approaches in sarcasm detection have neglected the intrinsic vagueness and uncertainty of human language in emotional expression and understanding. To address this gap, we propose a complex-valued fuzzy network by leveraging the mathematical formalisms of quantum theory and fuzzy logic. In particular, the target utterance to be recognized is considered as a quantum superposition of a set of separate words. The contextual interaction between adjacent utterances is described as the interaction between a quantum system and its surrounding environment, constructing the quantum composite system, where the weight of interaction is determined by a fuzzy membership function. In order to model both the vagueness and uncertainty, the aforementioned superposition and composite systems are mathematically encapsulated in a density matrix. Finally, a quantum fuzzy measurement is performed on the density matrix of each utterance to yield the probabilistic outcomes of sarcasm recognition. Extensive experiments are conducted on the MUStARD and the 2020 sarcasm detection Reddit track datasets, and the results show that our model outperforms a wide range of strong baselines.