  This thesis explores the use of entangled states in quantum computation and
quantum information science. Entanglement, a quantum phenomenon with no
classical counterpart, has been identified as an important and quantifiable
resource in many areas of theoretical quantum information science, including
quantum error correction, quantum cryptography, and quantum algorithms. We
first investigate the equivalence classes of a particular class of entangled
states (known as graph states due to their association with mathematical
graphs) under local operations. We prove that for graph states corresponding to
graphs with neither cycles of length 3 nor 4, the equivalence classes can be
characterized in a very simple way. We also present software for analyzing and
manipulating graph states. We then study quantum error-correcting codes whose
codewords are highly entangled states. An important area of investigation
concerning QECCs is to determine which resources are necessary in order to
carry out any computation on the code to an arbitrary degree of accuracy, while
simultaneously maintaining a high degree of resistance to noise. We prove that
transversal gates, which are designed to prevent the propagation of errors
through a system, are insufficient to achieve universal computation on almost
all QECCs. Finally, we study the problem of creating efficient quantum circuits
for creating entangling measurements. Entangling measurements can be used to
harness the apparent extra computing power of quantum systems by allowing us to
extract information about the global, collective properties of a quantum state
using local measurements. We construct explicit quantum circuits that create
entangling measurements, and show that these circuits scale polynomially in the
input parameters.
