This study proposes a generalized entropic uncertainty relation (EUR) for arbitrary multiple-observable in multipartite systems, and further derives a tighter lower bound by considering Holevo quality and mutual information. The research investigates the properties of the EUR and quantum coherence in the context of Schwarzschild space-time. It is found that Hawking radiation damages the coherence of the physically accessible region and increases the uncertainty. The properties of uncertainty in Schwarzschild space-time are explained from the systems' purity and information redistribution of different regions. The findings provide generalized EURs in multipartite systems, which may help understand quantumness and information paradox of black holes. The EUR is derived for m-measurements in n-party systems, and a stronger EUR is obtained by considering mutual information and Holevo quantity. The study also examines the vacuum structure of Dirac particles in Schwarzschild black holes, and analyzes the entropic uncertainty, quantum coherence, and mutual information in Schwarzschild black holes. The results show that Hawking radiation breaks quantum coherence and increases uncertainty. The uncertainty and coherence exhibit an anti-correlation with the Hawking temperature. As the Hawking temperature increases, the mutual information of systems outside the black hole's event horizon decreases, while that within the event horizon increases, indicating information redistribution from the physically accessible region to the physically inaccessible region. The findings suggest that the Hawking effect leads to the redistribution of information, which can explain the uncertainty dynamics of the Schwarzschild black hole. The study is supported by three-qubit W-like and Werner states in the background of Schwarzschild spacetime.This study proposes a generalized entropic uncertainty relation (EUR) for arbitrary multiple-observable in multipartite systems, and further derives a tighter lower bound by considering Holevo quality and mutual information. The research investigates the properties of the EUR and quantum coherence in the context of Schwarzschild space-time. It is found that Hawking radiation damages the coherence of the physically accessible region and increases the uncertainty. The properties of uncertainty in Schwarzschild space-time are explained from the systems' purity and information redistribution of different regions. The findings provide generalized EURs in multipartite systems, which may help understand quantumness and information paradox of black holes. The EUR is derived for m-measurements in n-party systems, and a stronger EUR is obtained by considering mutual information and Holevo quantity. The study also examines the vacuum structure of Dirac particles in Schwarzschild black holes, and analyzes the entropic uncertainty, quantum coherence, and mutual information in Schwarzschild black holes. The results show that Hawking radiation breaks quantum coherence and increases uncertainty. The uncertainty and coherence exhibit an anti-correlation with the Hawking temperature. As the Hawking temperature increases, the mutual information of systems outside the black hole's event horizon decreases, while that within the event horizon increases, indicating information redistribution from the physically accessible region to the physically inaccessible region. The findings suggest that the Hawking effect leads to the redistribution of information, which can explain the uncertainty dynamics of the Schwarzschild black hole. The study is supported by three-qubit W-like and Werner states in the background of Schwarzschild spacetime.