The k-spectrum of a string is the set of all distinct substrings of length k occurring in the string. K-spectra have many applications in bioinformatics including pseudoalignment and genome assembly. The Spectral Burrows-Wheeler Transform (SBWT) has been recently introduced as an algorithmic tool to efficiently represent and query these objects. The longest common prefix (LCP) array for a k-spectrum is an array of length n that stores the length of the longest common prefix of adjacent k-mers as they occur in lexicographical order. The LCP array has at least two important applications, namely to accelerate pseudoalignment algorithms using the SBWT and to allow simulation of variable-order de Bruijn graphs within the SBWT framework. In this paper we explore algorithms to compute the LCP array efficiently from the SBWT representation of the k-spectrum. Starting with a straightforward O(nk) time algorithm, we describe algorithms that are efficient in both theory and practice. We show that the LCP array can be computed in optimal O(n) time, where n is the length of the SBWT of the spectrum. In practical genomics scenarios, we show that this theoretically optimal algorithm is indeed practical, but is often outperformed on smaller values of k by an asymptotically suboptimal algorithm that interacts better with the CPU cache. Our algorithms share some features with both classical Burrows-Wheeler inversion algorithms and LCP array construction algorithms for suffix arrays.