This paper considers the recovery of higher-order tensor with intrinsic low-dimensional structure from quantized measurements. By introducing the low CANDE-COMP/PARAFAC (CP) rank constraint, we propose nonconvex models for both the general tensor recovery and the recovery of the tensors with tensor singular value decomposition (TSVD). We prove that the recovery errors for both optimization models go to zero when the dimension lengths of tensors go to infinity, and tensors with TSVD can theoretically reach a lower error. This paper also establishes a lower bound for any tensor recovery algorithm. Subsequently, a tensor-based alternating proximal gradient descent algorithm (TBAPGD) and a TSVD-based projected gradient descent algorithm (TSVD-PGD) are proposed to solve the nonconvex optimization problems. We provide a convergence guarantee for the former algorithm, and demonstrate the effectiveness of the latter through simulations. We empirically extend both algorithms to scenarios of missing data and without quantization rule information. Finally, we present experimental results on both synthetic data and real datasets to demonstrate the effectiveness and efficiency of the proposed methods.