I'm experiencing sharp pain that starts in my left arm then moves to my chest whenever I'm lying down, what could this be?

See your doctor. While this may be a musculoskeletal cause, any pain that radiates to the chest area should immediately be evaluated by a medical professional to rule out any cardiac causes.
More questions. Need to know character, duration, quality, aggravating, relieving factors, associated symptoms. Etc. Could be pinched nerve anywhere from spine to arm. Also other structures like joint etc need to be examined.