The purpose of this study is to see if giving participants quisinostat will prevent participants' uveal melanoma tumor from spreading. The researchers want to find out the effects that quisinostat has on participants' condition.

Inclusion Criteria:

1. Primary diagnosis of uveal melanoma (UM) with a lesion of at least 12 mm in largest basal diameter (LBD) as clinically determined by the treating Investigator. Cytologic determination of diagnosis is not required. Size is based on clinical assessment (e.g., by ultrasound or direct ophthalmoscopy) prior to enucleation or radiation therapy.
2. Definitive therapy of the primary UM must have been completed within 183 days of initiating protocol therapy.
3. High-risk (class 2) UM as determined by gene expression profiling (GEP; DecisionDx-UM, Castle Biosciences Inc., Friendswood, TX).
4. No evidence of metastatic disease.
5. Patients aged >18 years.
6. Eastern Cooperative Oncology Group (ECOG) performance status 0-1.
7. Life expectancy of greater than 3 months.
8. Ability to swallow and retain orally administered medication and no clinically significant gastrointestinal abnormalities that may alter absorption, such as malabsorption syndrome or major resection of the stomach or bowels.
9. Adequate organ and marrow function as defined by the local institutional lab and treating physician.
10. Women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation until 6 months after completion of quisinostat administration. Women of childbearing potential must have a negative urine or serum pregnancy test within 14 days prior to study entry.
11. Ability to understand and the willingness to sign a written informed consent document.

Exclusion Criteria:

1. Additional malignancy that is progressing or requires active treatment. Exceptions include the following cancers: basal cell carcinoma or squamous cell carcinoma of the skin that has undergone potentially curative therapy, in situ cervical cancer, ductal carcinoma in situ (DCIS), incidentally discovered asymptomatic thyroid cancer, elevated levels of prostate-specific antigen (PSA) stable on hormonal therapy with no otherwise detectable disease, and a previous diagnosis of malignancy that has shown no evidence of disease progression for 2 years or longer.
2. Any major surgery or extensive radiotherapy except that which is required for definitive treatment of primary UM.
3. Previous adjuvant treatment for UM after definitive primary tumor therapy.
4. History of prior Histone Deacetylase (HDAC) inhibitor use.
5. Patients that cannot be taken off medications that are potent inhibitors of cytochrome (CYP) 3a4/A5 (CYP3a4/A5) and CYP2C9. Inclusion of these patients and of patients on warfarin will require discussion and approval by the Sponsor-Investigator prior to enrollment.
6. Use of other investigational drugs within 28 days or five half-lives, whichever is shorter, with a minimum of 14 days from the last dose preceding the first dose of study treatment and during the study.
7. Known immediate or delayed hypersensitivity reaction or idiosyncrasy to drugs chemically related to quisinostat.
8. A QT interval corrected for heart rate using the Bazett's formula (QTcB) ≥ 480 msec or history of long QT syndrome.
9. Known Human Immunodeficiency Virus (HIV), Hepatitis B Virus (HBV), or Hepatitis C Virus (HCV) infection except for patients with cleared HBV and HCV infection demonstrated by undetectable viral levels by polymerase chain reaction (PCR). HIV-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with quisinostat.
10. Patients with a cardiac ejection fraction outside of the normal range as defined by institutional standards or with a history of clinically significant cardiac arrhythmia as determined by a cardiologist.
11. Uncontrolled intercurrent illness including but not limited to ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, New York Heart Association (NYHA) Classifications 2-4, or psychiatric illness/social situations that would limit compliance with study requirements.
12. History or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the patient's participation for the full duration of the trial, or that makes participation in the trial to be not in the best interest of the patient in the opinion of the treating Investigator.
13. Known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial.
14. Impaired decision-making capacity.