When you file Married Filing Separately, you are not eligible for most tax credits. The government does this to discourage people from claiming tax credits for which they would not be eligible if they filed jointly. Earned Income Credit is one of the credits you are not eligible for under that filing status.
If you have a qualifying child and you lived apart from your spouse for the last six months of the year , then you can file as Head of Household and qualify for EIC and other credits. This is the only circumstance in which you can file other than Married Filing Jointly or Married Filing Separately if you are legally married.
More information about these filing statuses is in the link below.