In this paper, an entropy-based quantum neuro-fuzzy inference system (EQNFIS) for classification applications is proposed. The EQNFIS model is a five-layer structure, which combines the traditional Takagi-Sugeno-Kang (TSK). Layer 2 of the EQNFIS model contains quantum membership functions, which are multilevel activation functions. Each quantum membership function is composed of the sum of sigmoid functions shifted by quantum intervals. A self-constructing learning algorithm, which consists of the self-clustering algorithm (SCA), quantum fuzzy entropy, and the backpropagation algorithm, is also proposed. The proposed SCA method is a fast, one-pass algorithm that dynamically estimates the number of clusters in an input data space. Quantum fuzzy entropy is employed to evaluate the information on pattern distribution in the pattern space. With this information, we can determine the number of quantum levels. The backpropagation algorithm is used to tune the adjustable parameters. Simulations were conducted to show the performance and applicability of the proposed model.