  Non-locality without inequality is an elegant argument introduced by L. Hardy
for two qubit systems, and later generalised to $n$ qubits, to establish
contradiction of quantum theory with local realism. Interestingly, for $n=2$
this argument is actually a corollary of Bell-type inequalities, viz. the
CH-Hardy inequality involving Bell correlations, but for $n$ greater than 2 it
involves $n$-particle probabilities more general than Bell-correlations. In
this paper, we first derive a chain of completely new local realistic
inequalities involving joint probabilities for $n$ qubits, and then, associated
to each such inequality, we provide a new Hardy-type local reality constraint
without inequalities. Quantum mechanical maximal violations of the chain of
inequalities and of the associated constraints are also studied by deriving
appropriate Cirel'son type theorems. These results involving joint
probabilities more general than Bell correlations are expected to provide a new
systematic tool to investigate entanglement.
