*Snacks and drinks will served at 4 pm*
One important application of quantum computers is efficiently simulating many-body quantum systems. The quantum computing equivalent of the vast array of diagnostic tools that extract information from classical numerical simulation are still being developed. Along these lines, we present a quantum algorithm to compute the entanglement spectrum of arbitrary quantum states. The interesting universal part of the entanglement spectrum is typically contained in the largest eigenvalues of the density matrix which can be obtained from the lower Renyi entropies through the Newton-Girard method. Obtaining the $p$ largest eigenvalues ($\lambda_1>\lambda_2… >\lambda_p$) requires a parallel circuit depth of $O(p(\lambda_1/\lambda_p)^p)$ and $O(p\log(N))$ qubits where up to $p$ copies of the quantum state defined on a Hilbert space of size $N$ are needed as the input. We validate this procedure for the entanglement spectrum of the topologically-ordered Laughlin wave function corresponding to the quantum Hall state at filling factor 1/3. Our scaling analysis exposes the tradeoffs between time and number of qubits for obtaining the entanglement spectrum in the thermodynamic limit using finite-size digital quantum computers. Importantly, we will also present results from implementing this algorithm on a digital quantum computing platform of trapped ion qubits to extract the second Renyi entropy of the ground state of a 2-site Fermi-Hubbard model.