Relation between quantum entanglement and quantum state complexity Both quantum entanglement and quantum state complexity are important in quantum information processing. They are usually highly correlated, i.e., roughly a state with a higher entanglement corresponds to a higher quantum state complexity or a complex state is usually highly entangled. But of course this correspondence is not exact. There are some highly entangled states that are not complex quantum states, for example quantum states represented by branching MERA.
On the other hand, if we use the geometric measure of entanglement (defined as the minimal distance to the nearest separable state w.r.t. a certain distance metric in Hilbert space) to justify the entanglement, then it seems it's very similar with the definition of quantum state complexity(the minimal distance to a simple product state). If we only consider pure states and choose the same distance metric for them, for example the Fubini-Study distance or Bures distance, then they are really almost identical. 
Of course, when we are talking about state complexity, it's better to use the more physically motivated 'quantum circuit complexity' to measure the distance. But still this distance can also be used to define the geometric measure of entanglement(maybe it's not a perfect distance measure for entanglement).
Then what's the relationship between entanglement and quantum state complexity? Are they essentially two different distance measures on Hilbert space?  What should be the optimal metrics for them?
Or, if entanglement and complexity are both distance measures on the Hilbert space, can we find a transformation between these two metrics? 
 A: As you noted, it is true that there is some coincidence on geometric entanglement and quantum circuit complexity. But there are also several counterexamples so we should distinguish between them:
1) Haar random states and random physical states: it is known that true random state in a Hilbert space of n-qubits cannot be attainable from a product state with poly(N) gates (https://arxiv.org/abs/1102.1360). However, using O(N^3) random two-qubit gates, you can make a quantum state with such a large entanglement (https://arxiv.org/abs/1109.4391). Even though these studies used (Reyni) entanglement entropy for the entanglement measures, I am sure that geometric entanglement also shows similar behavior. Indeed, I have some numerical evidence for that (you may see e.g. https://doi.org/10.1103/PhysRevA.93.042314). So entanglement and circuit depth are different. You may also see https://arxiv.org/abs/1310.2702 in this context.
2) After applying $N$ random 2-qubit gates sequentially to a product state of $N$ qubits, the maximum overlap to a product states $e^{-aN}$. This also contrasts to the GHZ type of states that you need $N$ gates to prepare, but the maximum overlap to a product state is given by a constant.
So when they behave similarly? For a shallow depth (O(N^2)) quantum circuit with random unitary two-qubit gates, geometric entanglement grows as the number of gates increases. When the gates are special (such as CNOT) or the number of gates are too large, this coincidence breaks down.
