To meet the required huge data analysis, organization and storage demand, hashing technique has got a lot of attention as it aims to learn efficient binary representation from the original high-dimensional data. In this paper, we focus on the unsupervised spectral hashing due to its effective manifold embedding. Existing spectral hashing methods mainly suffer from two problems, i.e., the inefficient spectral candidate and intractable binary constraint for spectral analysis. To overcome these two problems, we propose to employ spectral rotation to seek better spectral solution and adopt the alternating projection algorithm to settle the complex code constraints, which are therefore named as Spectral Hashing with Spectral Rotation (SHSR) and Alternating Discrete Spectral Hashing (ADSH), respectively. To enjoy the merits of both methods, the spectral rotation technique is finally combined with the original spectral objective, which aims to simultaneously learn better spectral solution and more efficient discrete codes and is called as Discrete Spectral Hashing (DSH). Further, efficient optimization algorithms are also provided, which just take comparable time complexity to existing hashing methods. To evaluate the proposed three methods, extensive comparison experiments and studies are conducted on four large-scale datasets for the image retrieval task, and the noticeable performance beats several state-of-theart spectral hashing methods on different evaluation metrics. IEEE