Contact tracing via smartphone applications is expected to be of major importance for maintaining control of the COVID-19 pandemic. However, viable deployment demands a minimal quarantine burden on the general public. That is, consideration must be given to unnecessary quarantining imposed by a contact tracing policy. Previous studies have modeled the role of contact tracing, but have not addressed how to balance these two competing needs. We propose a modeling framework that captures contact heterogeneity. This allows contact prioritization: contacts are only notified if they were acutely exposed to individuals who eventually tested positive. The framework thus allows us to address the delicate balance of preventing disease spread while minimizing the social and economic burdens of quarantine. This optimal contact tracing strategy is studied as a function of limitations in testing resources, partial technology adoption, and other intervention methods such as social distancing and lockdown measures. The framework is globally applicable, as the distribution describing contact heterogeneity is directly adaptable to any digital tracing implementation.