  Determining the relationship between quantum correlation sets is a
long-standing open problem. The most well-studied part of the hierarchy is
captured by the chain of inclusions $\mathcal C_q \subseteq \mathcal C_{qs}
\subsetneq \mathcal C_{qa} \subseteq \mathcal C_{qc}$. The separation $\mathcal
C_{qs} \neq \mathcal C_{qa}$, showing that the set of quantum spatial
correlations is not closed, was proven in breakthrough work by Slofstra
[arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)]. Resolving the question of
$\mathcal C_{qa} = \mathcal C_{qc}$ would resolve the Connes Embedding
Conjecture and would represent major progress in the mathematical field of
operator algebras. In this work, we resolve the ambiguity in the first
inclusion, showing that $\mathcal{C}_q \neq \mathcal{C}_{qs}$. We provide an
explicit construction of a correlation that can be attained on a tensor product
of infinite-dimensional Hilbert spaces but not finite-dimensional ones. This
property is also conjectured to be possessed by any correlation which maximally
violates the $I_{3322}$ inequality.
