Protein secondary structure is crucial to creating an information bridge between the primary and tertiary (3D) structures. Precise prediction of eight-state protein secondary structure (PSS) has significantly utilized in the structural and functional analysis of proteins in bioinformatics. Deep learning techniques have been recently applied in this research area and raised the eight-state (Q8) protein secondary structure prediction accuracy remarkably. Nevertheless, from a theoretical standpoint, there are still lots of rooms for improvement, specifically in the eight-state PSS prediction. In this study, we have presented a new deep convolutional neural network (DCNN), namely PS8-Net, to enhance the accuracy of eight-class PSS prediction. The input of this architecture is a carefully constructed feature matrix from the proteins sequence features and profile features. We introduce a new PS8 module in the network, which is applied with skip connection to extracting the long-term inter-dependencies from higher layers, obtaining local contexts in earlier layers, and achieving global information during secondary structure prediction. Our proposed PS8-Net achieves 76.89%, 71.94%, 76.86%, and 75.26% Q8 accuracy respectively on benchmark CullPdb6133, CB513, CASP10, and CASP11 datasets. This architecture enables the efficient processing of local and global interdependencies between amino acids to make an accurate prediction of each class. To the best of our knowledge, PS8-Net experiment results demonstrate that it outperforms all the state-of-the-art methods on the aforementioned benchmark datasets.