Hyperspectral image (HSI) classification aims at assigning a unique label for every pixel to identify categories of different land covers. Existing deep learning models for HSIs are usually performed in a traditional learning paradigm. Being emerging machines, quantum computers are limited in the noisy intermediate-scale quantum (NISQ) era. The quantum theory offers a new paradigm for designing deep learning models. Motivated by the quantum circuit (QC) model, we propose a quantum-inspired spectral-spatial network (QSSN) for HSI feature extraction. The proposed QSSN consists of a phase-prediction module (PPM) and a measurement-like fusion module (MFM) inspired from quantum theory to dynamically fuse spectral and spatial information. Specifically, QSSN uses a quantum representation to represent an HSI cuboid and extracts joint spectral-spatial features using MFM. An HSI cuboid and its phases predicted by PPM are used in the quantum representation. Using QSSN as the building block, we further propose an end-to-end quantum-inspired spectral-spatial pyramid network (QSSPN) for HSI feature extraction and classification. In this pyramid framework, QSSPN progressively learns feature representations by cascading QSSN blocks and performs classification with a softmax classifier. It is the first attempt to introduce quantum theory in HSI processing model design. Substantial experiments are conducted on three HSI datasets to verify the superiority of the proposed QSSPN framework over the state-of-the-art methods.