The study of ground spaces of local Hamiltonians is a fundamental task in condensed matter physics. In terms of computational complexity theory, a common focus has been to estimate a given Hamiltonian’s ground state energy. However, from a physics perspective, it is often more relevant to understand the structure of the ground space itself. In this paper, we pursue this latter direction by introducing the physically well-motivated notion of “ground state connectivity” of local Hamiltonians, which captures problems in areas ranging from stabilizer codes to quantum memories. We show that determining how “connected” the ground space of a local Hamiltonian is can range from QCMA-complete to NEXPcomplete. As a result, we obtain a natural QCMA-complete problem, a goal which has proven elusive since the conception of QCMA over a decade ago. Our proofs crucially rely on a new technical tool, the Traversal Lemma, which analyzes the Hilbert space a local unitary evolution must traverse under certain conditions, and which we believe may be of independent interest. [A technical draft of this paper can be found at: http://arxiv.org/abs/1409.3182] Over the last fifteen years, the merging of condensed matter physics and computational complexity theory has given rise to a new field of study known as quantum Hamiltonian complexity [Osb12, GHL14]. The cornerstone of this field is arguably Kitaev’s [KSV02] quantum version of the Cook-Levin theorem [Coo72, Lev73], which says that the problem of estimating the ground state energy of a local Hamiltonian is complete for the class Quantum Merlin Arthur (QMA), where QMA is a natural generalization of NP. Here, a k-local Hamiltonian is an operator H = ∑i Hi acting on n qubits, such that each local Hermitian constraint Hi acts non-trivially on k qubits. The ground state energy of H is simply the smallest eigenvalue of H, and the corresponding eigenspace is known as the ground space of H. Kitaev’s result spurred a long line of subsequent works on variants of the ground energy estimation problem (see, e.g. [Osb12, GHL14] for surveys), known as the k-local Hamiltonian problem (k-LH). For example, Oliveira and Terhal showed that LH remains QMA-complete in the physically motivated case of qubits arranged on a 2D lattice [OT08]. Bravyi and Vyalyi [BV05] proved that the commuting variant of 2-LH is in NP [BV05]. More recently, the complexity of 2-LH was completely characterized by Cubitt and Montanaro [CM13] in a manner analogous to Schaeffer’s dichotomy theorem for Boolean satisfiability [Sch78]. Thus, k-LH has served as an excellent “benchmark” problem for delving into the complexity of problems encountered in condensed matter physics. Yet, physically speaking, what is often more relevant than the ground state energy is an understanding of the ground space itself. What are its properties? For example, is it topologically ordered? Can we evaluate local observables against it [Osb12]? It is this direction which we pursue in this paper. Specifically, in this paper we define a notion of connectivity of the ground space of H, which roughly asks: Given ground states |ψ〉 and |φ〉 of H as input, are they “connected” through the ground space of H? Somewhat more formally, we have (see Section 2 of technical draft for a formal definition): Definition 1 (Ground State Connectivity (GSCON) (informal)). Given as input a local Hamiltonian H and ground states |ψ〉 and |φ〉 of H (specified via quantum circuits), as well as parameters m and l, does there exist a sequence of l-qubit unitaries (Ui) i=1 such that: 1. (|ψ〉 mapped to |φ〉) Um · · ·U1 |ψ〉 ≈ |φ〉, and 2. (intermediate states in ground space) ∀ i ∈ [m], Ui · · ·U1 |ψ〉 is in the ground space of H? In other words, GSCON asks whether there exists a sequence of m unitaries, each acting on (at most) l qubits, mapping the initial state |ψ〉 to the final state |φ〉 through the ground space of H. We stress that the parameters m (i.e. number of unitaries) and l (i.e. the locality of each unitary) are key; as we discuss shortly, depending on their setting, the complexity of GSCON can vary greatly. ∗Department of Computer Science, Virginia Commonwealth University, USA. †Centre for Quantum Technologies, National University of Singapore, Singapore.