Presuming that there aren't nonlocal constraints, a differential operator that is polynomial in differential operators is local, it doesn't have to be quadratic. My understanding is that irrational or transcendental functions of differential operators are generally nonlocal (though that's perhaps a Question for math.SE).
A given space of solutions implies a particular nonlocal choice of boundary conditions, unless the equations are on a compact manifold (which, however, is itself a nonlocal structure). There is always an element of nonlocality when we discuss solutions in contrast to equations.
[For the anti-locality of the operator $(-\nabla^2+m^2)^\lambda$ for odd dimension and non-integer $\lambda$, one can see I.E. Segal, R.W. Goodman, J. Math. Mech. 14 (1965) 629 (for a review of this paper, see here).]
EDIT: Sorry, I should have gone straight to Hegerfeldt's theorem. Schrodinger's equation is enough like the heat equation to be nonlocal in Hegerfeldt's sense. There are two theorems, from 1974 in PRD and from 1994 in PRL, but in arXiv:quant-ph/9809030 we have, of course with references to the originals,
Theorem 1. Consider a free relativistic particle of positive or
zero mass and arbitrary spin. Assume that at time $t=0$ the particle
is localized with probability 1 in a bounded region V . Then there is
a nonzero probability of finding the particle arbitrarily far away at
any later time.
Theorem 2. Let the operator $H$ be self-adjoint and bounded from below.
Let $\mathcal{O}$ be any operator satisfying $$0\le \mathcal{O} \le \mathrm{const.}$$ Let
$\psi_0$ be any vector and define $$\psi_t \equiv \mathrm{e}^{-\mathrm{i}Ht}\psi_0.$$ Then one of the following two
alternatives holds. (i) $\left<\psi_t,\mathcal{O}\psi_t\right>\not=0$ for almost
all $t$ (and the set of such t's is dense and open) (ii)
$\left<\psi_t,\mathcal{O}\psi_t\right>\equiv 0$ for all $t$.
Exactly how to understand Hegerfeldt's theorem is another question. It seems almost as if it isn't mentioned because it's so inconvenient (the second theorem, in particular, has a rather simple statement with rather general conditions), but a lot depends on how we define local and nonlocal.
I usually take Hegerfeldt's theorem to be a non-relativistic cognate of the Reeh-Schlieder theorem in axiomatic QFT, although that's perhaps heterodox, where microcausality is close to the only definition of local. Microcausality is one of the axioms that leads to the Reeh-Schlieder theorem, so, no nonlocality.
The locality of a QFT refers to the operator algebra. The (non-) locality of Bell's theorem refers to the states (rays) of the Hilbert space. These are different notions of locality, and they coexist peacefully.
To quote, Fredenhagen1
Apart from these problems, there is a deeper reason why it is fortunate to separate the construction of observables from the construction of states. This is the apparent conflict between the principle of locality, which in particular governs classical field theory, and the existence of nonclassical correlations (entanglement) in quantum systems, often referred to as non-locality of quantum physics. As a matter of fact it turns out that the algebra of observables is completely compatible with the locality principle whereas the states typically exhibit nonlocal correlations.
--
1 An Introduction to Algebraic Quantum Field Theory, from the book Advances in Algebraic Quantum Field Theory.
Best Answer
The short answer is yes, in quantum mechanics quantum non-locality refers to the apparent instantaneous propagation of correlations between entangled systems, irrespective of their spatial separation. In quantum field theory, the notion of locality may have a different meaning, as pointed out already in a comment.
Details: The notions of locality and non-locality in Quantum Mechanics have been originally defined in the context of the EPR controversy between Einstein and Bohr on the phenomenon of quantum entanglement.
Basically, the general "principle of locality" (Wikipedia ref.) requires that "for an action at one point to have an influence at another point, something in the space between the points, such as a field, must mediate the action". In view of the theory of relativity, the speed at which such an action, interaction, or influence can be transmitted between distant points in space cannot exceed the speed of light. This formulation is also known as "Einstein locality" or "local relativistic causality". It is often stated as "nothing can propagate faster than light, be it energy or merely information" or simply "no spooky action-at-a-distance", as Einstein himself put it. For the past 20 years or so it has been referred to also as the "no-signaling" condition.
The phenomenon of entanglement between quantum systems raised the non-locality problem first noted in the EPR paper: A projective measurement on a quantum system at one space location instantly collapses the state of an entangled counterpart at a distant location. Quantum mechanical non-locality refers to this apparent entanglement-mediated violation of Einstein locality.
The remarkable thing about quantum non-locality, however, is that it actually does not imply violation of relativistic causality. Although entanglement correlations are affected instantaneously, they cannot be harnessed for faster-than-light communications. The reason is that the outcome of the local projective measurement is itself statistic and cannot be predicted beforehand. If the same kind of measurement is performed on multiple copies of identically prepared pairs of entangled systems, the overall statistical result is that locally both systems conform to the statistics prescribed by their respective local quantum states. The "spooky action-at-a-distance" of the distant measurement gets wiped out in the total statistics.
There is already a host of related questions on Physics.SE. See for instance Quantum entanglement and spooky action at a distance and similar.