How is Brown vs. Plata related to the Eighth Amendment? How did Escobedo vs. Illinois affect interrogations in the United States?

The 8th Amendment pertains to prison overcrowding and the Brown vs. Plata case. First, the 8th Amendment protects citizens against cruel and unusual punishments. In 2011, the Supreme Court affirmed that the prison conditions in the state of California violated the cruel and unusual punishment clause in the 8th Amendment. In a 5-4 decision, the justices ordered California to reduce its inmate population by at least 46,000 inmates. According to a New York Times article, extreme overcrowding in California prisons has resulted in administrative negligence and high prisoner suicide rates. Prisoners with mental and physical health issues often received little care for their conditions.
Here's a concise summary of the Brown vs. Plata case.
In regards to the sentencing of a defendant, lawyers and/ or prosecutors may highlight aggravating and mitigating factors in a bid to influence the judge's sentencing decision. Mitigating factors are used by the defendant's lawyers to plead for leniency. Some of these factors include the remorse of the defendant, the defendant's lack of a criminal record, the defendant's minor role in his crime, or the evidence of physical and/ or mental incapacity at the time of his crime.
On the other hand, prosecutors try to highlight aggravating factors that may lead to a harsher sentence for the defendant. Some of these factors include the severity of the victim's suffering, the level of the victim's vulnerability, the defendant's record of similar convictions, and the level of the defendant's participation in his crime.
In the case of interrogations, coerced confessions violate the Fifth Amendment right against self-incrimination. In Escobedo vs. Illinois, Escobedo was allegedly denied his Sixth Amendment right to counsel; therefore, he had no way of protecting his Fifth Amendment rights. Before the advent of the Miranda rule (where defendants are advised of their right to remain silent and/ or to have a lawyer present during an interrogation), the totality of circumstances test was used to determine whether a defendant's confession was in any way coerced (in violation of that person's Fifth Amendment rights).
In Rhode Island vs. Innis, the Supreme Court concluded that Thomas J. Innis's Fifth Amendment rights were not violated. Accordingly, Innis was arrested for committing a robbery with a sawed-off shotgun. At the time of his arrest, the shotgun was nowhere to be found. On the way to the police station, the officers openly admitted their fears that the shotgun might be found by children from a nearby school for the disabled. In response to this conversation, Innis offered to show the officers where the shotgun was. In a  6-3 decision, the Supreme Court ruled that the officers' conversation did not qualify as words that would elicit a self-incriminating confession from Innis.
In Quarles vs. NY, the Supreme Court ruled that Quarles had not been denied his Miranda rights. When Quarles was apprehended by an officer, the officer asked him about the location of his gun. After the officer received an answer from Quarles, he proceeded to read Quarles his Miranda rights. The court maintained that the officer's request constituted a public safety exception to the Miranda rule. Essentially, the officer had the right to protect his person and ensure the safety of bystanders before he read Quarles his Miranda rights.
https://constitution.findlaw.com/amendment5/annotation09.html

https://www.justia.com/criminal/aggravating-mitigating-factors/