Many eukaryotic cells are able to detect chemical gradients by directly measuring spatial concentration differences. The precision of such gradient sensing is limited by fluctuations in the binding of diffusing particles to specific receptors on the cell surface. Here, we explore the physical limits of the spatial sensing mechanism by modeling the chemotactic cell as an Ising spin chain subject to a spatially varying field. Our results demonstrate that the accuracy to sense the gradient direction not only increases dramatically with the cell size but also can be improved significantly by introducing receptor cooperativity. Thus, receptor coupling may open the possibility for small bacteria to perform spatial measurements of gradients, as supported by a recent experimental finding.