  Quantum theory and relativity are the pillar theories on which our
understanding of physics is based. Poincar\'e invariance is a fundamental
physical principle stating that the experimental results must be the same in
all inertial reference frames in Minkowski spacetime. It is a basic condition
imposed on quantum theory in order to construct quantum field theories, hence,
it plays a fundamental role in the standard model of particle physics too. As
is well known, Minkowski spacetime follows from clear physical principles, like
the relativity principle and the invariance of the speed of light. Here, we
reproduce such a derivation, but leave the number of spatial dimensions $n$ as
a free variable. Then, assuming that spacetime is Minkowski in $1+n$ dimensions
and within the framework of general probabilistic theories, we reconstruct the
qubit Bloch ball and finite dimensional quantum theory, and obtain that the
number of spatial dimensions must be $n=3$, from Poincar\'e invariance and
other physical postulates. Our results suggest a fundamental physical
connection between spacetime and quantum theory.
