  The complexity of a quantum state may be closely related to the usefulness of
the state for quantum computation. We discuss this link using the tree size of
a multiqubit state, a complexity measure that has two noticeable (and, so far,
unique) features: it is in principle computable, and non-trivial lower bounds
can be obtained, hence identifying truly complex states. In this paper, we
first review the definition of tree size, together with known results on the
most complex three and four qubits states. Moving to the multiqubit case, we
revisit a mathematical theorem for proving a lower bound on tree size that
scales superpolynomially in the number of qubits. Next, states with
superpolynomial tree size, the Immanant states, the Deutsch-Jozsa states, the
Shor's states and the subgroup states, are described. We show that the
universal resource state for measurement based quantum computation, the
2D-cluster state, has superpolynomial tree size. Moreover, we show how the
complexity of subgroup states and the 2D cluster state can be verified
efficiently. The question of how tree size is related to the speed up achieved
in quantum computation is also addressed. We show that superpolynomial tree
size of the resource state is essential for measurement based quantum
computation. The necessary role of large tree size in the circuit model of
quantum computation is still a conjecture; and we prove a weaker version of the
conjecture.
