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Semi-supervised segmentation remains challenging in medical imaging since the amount of annotated medical data is often scarce and there are many blurred pixels near the adhesive edges or in the low-contrast regions. To address the issues, we advocate to firstly constrain the consistency of pixels with and without strong perturbations to apply a sufficient smoothness constraint and further encourage the class-level separation to exploit the low-entropy regularization for the model training. Particularly, in this paper, we propose the SS-Net for semi-supervised medical image segmentation tasks, via exploring the pixel-level smoothness and inter-class separation at the same time. The pixel-level smoothness forces the model to generate invariant results under adversarial perturbations. Meanwhile, the inter-class separation encourages individual class features should approach their corresponding high-quality prototypes, in order to make each class distribution compact and separate different classes. We evaluated our SS-Net against five recent methods on the public LA and ACDC datasets. Extensive experimental results under two semi-supervised settings demonstrate the superiority of our proposed SS-Net model, achieving new state-of-the-art (SOTA) performance on both datasets. The code is available at https://github.com/ycwu1997/SS-Net.