Conversational sentiment analysis is an emerging, yet challenging Artificial Intelligence (AI) subtask. It aims to discover the affective state of each participant in a conversation. There exists a wealth of interaction information that affects the sentiment of speakers. However, the existing sentiment analysis approaches are insufficient in dealing with this task due to ignoring the interactions and dependency relationships between utterances. In this paper, we aim to address this issue by modeling intrautterance and inter-utterance interaction dynamics. We propose an approach called quantum-inspired interactive networks (QIN), which leverages the mathematical formalism of quantum theory (QT) and the long short term memory (LSTM) network, to learn such interaction dynamics. Specifically, a density matrix based convolutional neural network (DM-CNN) is proposed to capture the interactions within each utterance (i.e., the correlations between words), and a strong-weak influence model inspired by quantum measurement theory is developed to learn the interactions between adjacent utterances (i.e., how one speaker influences another). Extensive experiments are conducted on the MELD and IEMOCAP datasets. The experimental results demonstrate the effectiveness of the QIN model.