  Quantum correlations and non-projective measurements underlie a plethora of
information-theoretic tasks, otherwise impossible in the classical world.
Existing schemes to certify such non-classical resources in a
device-independent manner require seed randomness, which is often costly and
vulnerable to loopholes, for choosing the local measurements performed on
different parts of a multipartite quantum system. In this letter, we propose
and experimentally implement a semi-device independent certification technique
for both quantum correlations and non-projective measurements without seed
randomness. Our test is semi-device independent in the sense that it requires
only prior knowledge of the dimensions of the parts. We experimentally show a
novel quantum advantage in correlated coin tossing by producing specific
correlated coins from pairs of photons entangled in their transverse spatial
modes. We establish the advantage by showing that the correlated coin obtained
from the entangled photons cannot be obtained from two 2-level classical
correlated coins. The quantum advantage requires performing qubit trine
positive operator-valued measures (POVMs) on each part of the entangled pair,
thus also certifying such POVMs in a semi-device-independent manner. This proof
of concept firmly establishes a new cost-effective certification technique for
both generating non-classical shared randomness and implementing non-classical
measurements, which will be important for future multi-party quantum
communications.
