  Quantum correlations, in particular those, which enable to violate a Bell
inequality \cite{BELL}, open a way to advantage in certain communication tasks.
However, the main difficulty in harnessing quantumness is its fragility to,
e.g, noise or loss of particles. We study the persistency of Bell correlations
of GHZ based mixtures and Dicke states. For the former, we consider quantum
communication complexity reduction (QCCR) scheme, and propose new Bell
inequalities (BIs), which can be used in that scheme for higher persistency in
the limit of large number of particles $N$. In case of Dicke states, we show
that persistency can reach $0.482N$, significantly more than reported in
previous studies.
