  Prophet inequalities and secretary problems have been extensively studied in
recent years due to their elegance, connections to online algorithms,
stochastic optimization, and mechanism design problems in game theoretic
settings. Rubinstein and Singla developed a notion of combinatorial prophet
inequalities in order to generalize the standard prophet inequality setting to
combinatorial valuation functions such as submodular and subadditive functions.
For non-negative submodular functions they demonstrated a constant factor
prophet inequality for matroid constraints. Along the way they showed a variant
of the correlation gap for non-negative submodular functions.
  In this paper we revisit their notion of correlation gap as well as the
standard notion of correlation gap and prove much tighter and cleaner bounds.
Via these bounds and other insights we obtain substantially improved constant
factor combinatorial prophet inequalities for both monotone and non-monotone
submodular functions over any constraint that admits an Online Contention
Resolution Scheme. In addition to improved bounds we describe efficient
polynomial-time algorithms that achieve these bounds.
