  Measurements in quantum theory exhibit incompatibility, i.e., they can fail
to be jointly measurable. An intuitive way to represent the (in)compatibility
relations among a set of measurements is via a hypergraph representing their
joint measurability structure: its vertices represent measurements and its
hyperedges represent (all and only) subsets of compatible measurements.
Projective measurements in quantum theory realize (all and only) joint
measurability structures that are graphs. On the other hand, general
measurements represented by positive operator-valued measures (POVMs) can
realize arbitrary joint measurability structures. Here we explore the scope of
joint measurability structures realizable with qubit POVMs. We develop a
technique that we term marginal surgery to obtain nontrivial joint
measurability structures starting from a set of compatible measurements. We
show explicit examples of marginal surgery on a special set of qubit POVMs to
construct joint measurability structures such as $N$-cycle and $N$-Specker
scenarios for any integer $N\geq 3$. We also show the realizability of various
joint measurability structures with $N\in\{4,5,6\}$ vertices. In particular, we
show that all possible joint measurability structures with $N=4$ vertices are
realizable. We conjecture that all joint measurability structures are
realizable with qubit POVMs. This contrasts with the unbounded dimension
required in R. Kunjwal et al., Phys. Rev. A 89, 052126 (2014). Our results also
render this previous construction maximally efficient in terms of the required
Hilbert space dimension. We also obtain a sufficient condition for the joint
measurability of any set of binary qubit POVMs which powers many of our results
and should be of independent interest.
