  We investigate the possibility of eavesdropping on a quantum key distribution
network by local sequential quantum unsharp measurement attacks by the
eavesdropper. In particular, we consider a pure two-qubit state shared between
two parties Alice and Bob, sharing quantum steerable correlations that form the
one-sided device-independent quantum key distribution network. One qubit of the
shared state is with Alice and the other one while going to the Bob's place is
intercepted by multiple sequential eavesdroppers who perform quantum unsharp
measurement attacks thus gaining some positive key rate while preserving the
quantum steerable correlations for the Bob. In this way, Bob will also have a
positive secret key rate although reduced. However, this reduction is not that
sharp and can be perceived due to decoherence and imperfection of the
measurement devices. At the end, we show that an unbounded number of
eavesdroppers can also get secret information in some specific scenario.
