Recent studies have demonstrated that coherent weak measurements allow Einstein-Podolsky-Rosen steering to be shared sequentially between a single Alice and multiple Bobs. Theoretically, this permits an unbounded sequence in the high-dimensional limit. In this work, we investigate whether the experimentally simpler strategy of probabilistic projective measurements (PPM) can achieve similar results. We identify a fundamental distinction between the two approaches: while coherent weak measurements perturb the quantum state quadratically $\left[\mathcal{O}\left({\eta }_k^2\right)\right]$, PPM induces an effective linear depolarizing contribution $\left[\mathcal{O}\left({\gamma }_k\right)\right]$. We prove that this linear scaling restricts the information-to-disturbance ratio to $\mathcal{O}\left(1\right)$, causing the quantum correlations to decay in an arithmetic progression. This imposes a strict bound on the number of accessible Bobs in finite-dimensional systems, severely limiting sequence lengths for qubits. However, we demonstrate that high-dimensional entanglement is robust enough to sustain the sequence, yielding an asymptotic sharing capacity of $N\sim d/lnd$. Ultimately, our results reveal a crucial physical trade-off: while PPM offers experimental simplicity, it consumes quantum correlations at a significantly higher rate than coherent weak measurements.

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• Quantum information theory
• Quantum measurements